76. Find the local maxima and local minima, if any, of the following functions.

       Find also the local maximum and the local minimum values, as the case may be:

(i) f(x) = x².

(i) we have, f(x) = x².

f(x) = 2x.

andf(x) = 2.

AR extreme point, f(x) = 0

2x = 0

x = 0.

When x = 0, f(0) = 2 > 0.

∴x = 0 is a point of local minima and value of local minimum is given by f(0) = 0² = 0.

(ii) g(x) = x³ 3x

A(ii)

we have, g(x) = x³- 3x

$\Rightarrow$ g'(x) = 3x²- 3

$\Rightarrow$ g''(x) = 6x.

At extreme point,

g'(x) = 0

3x²- 3 = 0.

3(x²- 1) = 0 ⇒ 3(x - 1)(x + 1) = 0.

x = 1 or x = -1.

At x = 1, g"(1) = 6.1 = 6 > 0.

So, x = 1 is a point of local minima and value of local minimum is given by g(1) = 1³- 3.1 = 1 - 3 = - 2.

And at x = -1, g"( -1) = 6 ( -1) = 6 < 0.

So, x = -1 is a point of local minima and value of local minimum is given by

g(- 1) = (- 1)³- 3(- 1) = 1

...more

(i) we have, f(x) = x².

f(x) = 2x.

andf(x) = 2.

AR extreme point, f(x) = 0

2x = 0

x = 0.

When x = 0, f(0) = 2 > 0.

∴x = 0 is a point of local minima and value of local minimum is given by f(0) = 0² = 0.

(ii) g(x) = x³ 3x

A(ii)

we have, g(x) = x³- 3x

$\Rightarrow$ g'(x) = 3x²- 3

$\Rightarrow$ g''(x) = 6x.

At extreme point,

g'(x) = 0

3x²- 3 = 0.

3(x²- 1) = 0 ⇒ 3(x - 1)(x + 1) = 0.

x = 1 or x = -1.

At x = 1, g"(1) = 6.1 = 6 > 0.

So, x = 1 is a point of local minima and value of local minimum is given by g(1) = 1³- 3.1 = 1 - 3 = - 2.

And at x = -1, g"( -1) = 6 ( -1) = 6 < 0.

So, x = -1 is a point of local minima and value of local minimum is given by

g(- 1) = (- 1)³- 3(- 1) = 1 + 3 = 2.

(iii) $h\left(x\right)=sinx+cosx.0<x<\frac{\pi }{2}$

A(iii)

we have, h(x) = sin x + cos x.

$\Rightarrow$ h'(x) = cos x - sin x

$\Rightarrow$ h'(x) = -sin x - cos x.

At extreme points,

h'(x) = 0

cosx - sin x = 0

cosx = sin x

$\Rightarrow \frac{cosx}{cosx}=\frac{sinx}{cosx}$

$\Rightarrow$ 1 = tan x

$\Rightarrow tanx=tan\frac{\pi }{4}$

$\Rightarrow x=\frac{\pi }{4}.$

(iv) $f\left(x\right)=sinx-cosx,0<x<2\pi$

A(iv)

we have, f(x) = sin x cos x.

$\Rightarrow$ f(x) = cos x + sin x

$\Rightarrow$ f(x) = sin x + cos x.

At extreme points,

f(x) = 0

$\Rightarrow$ cosx + sin x = 0.

$\Rightarrow \frac{sinx}{cosx}=\frac{-cosx}{cosx}$

$\Rightarrow tanx=-1=-tan\frac{\pi }{4}=tan\left(\pi -\frac{\pi }{4}\right)=tan\left(2\pi -\frac{\pi }{4}\right)$

$\therefore x=\left(\pi -\frac{\pi }{4}\right)$ IInd quadrate or $\left(2\pi -\frac{\pi }{4}\right)$ IIIth quadrate

$\Rightarrow x=\frac{4\pi -\pi }{4}\frac{8x-\pi }{4}$

$\Rightarrow x=\frac{3\pi }{4}orx=\frac{f\pi }{4}.$

$Atx=\frac{3\pi }{4},f^{\prime \prime }\left(\frac{3\pi }{4}\right)=-sin\frac{3\pi }{4}+cos\frac{3\pi }{4}$

$=-sin\left(\pi -\frac{\pi }{4}\right)+cos\left(\pi -\frac{\pi }{4}\right)$

$=-sin\frac{\pi }{4}-cot\frac{\pi }{4}$

(v) f(x) = x³- 6x² + 9x + 15.

A(v)

we have, f(x) = x³- 6x² + 9x + 15.

$\Rightarrow$ f(x) = 3x²- 12x + 9.

$\Rightarrow$ f'(x) = 6x - 12.

At extreme point, f'(x) = 0.

$\Rightarrow$ 3x²- 12x + 9 = 0.

$\Rightarrow$ x²- 4x + 3 = 0

$\Rightarrow$ x²-x - 3x + 3 = 0

$\Rightarrow$ x(x - 1) -3(x - 1) = 0

$\Rightarrow$ (x - 1)(x - 3) = 0

$\Rightarrow$ x = 1 or x = 3.

At, x = 1,f"(1) = 6 × 1 - 12 = 6 - 12 = - 6 < 0

∴x = 1 is a point of local maxima and the value of local

Maximum is given by f(1) = 1³- 6(1)² + 9(1) + 15

= 1 - 6 + 9 + 15

= 19.

And at x = 3,f"(3) = 6 × 3 - 12 = 18 - 12 = 6 > 0

∴x = 3 is a point of local minima and the value of

local minimum is given by f (3) = 3³- 6(3)² + 9(3) + 15.

= 27 - 54 + 27 + 15.

= 15.

(vi) $g(x)=\frac{x}{2}+\frac{2}{x},x>0$

A(vi)

We have, $g(x)=\frac{x}{2}+\frac{2}{x},x>0$

$\Rightarrow g^{\prime }(x)=\frac{1}{2}-\frac{2}{x^2}$

$\Rightarrow g{g}^{\prime z}(x)=\frac{x^2-4}{2x^2}=\frac{(x-2)(x+2)}{2x^2}$

$g^{\prime \prime }(x)=\frac{4}{x^3}.$

At, extreme point, g'(x) = 0

$\Rightarrow \frac{(x+2)(x-2)}{x^2}=0$

$\Rightarrow$ x = -2 or x = 2

Given, that x > 0, hence we have x = 2.

At, x = 2, ${\stackrel{?}{g}}^{\prime }(x)=\frac{4}{2^3}=\frac{4}{8}=\frac{1}{2}>0$

∴x = 2 is a point of local minima and value of local

Minimum is given by g(2) = $\frac{2}{2}+\frac{2}{2}=1+1=2.$

(vii) $f(x)=\frac{1}{x^2+2}$

A(vii)

Given, $f(x)=\frac{1}{x^2+2}$

$\Rightarrow g^{\prime }(x)=-\frac{1}{{\left(x^2+2\right)}^2}\frac{d}{dx}\left(x^2+2\right)=\frac{-2x}{{\left(x^2+2\right)}^2}.$

$\Rightarrow g^{\prime \prime }(x)=-\left[\frac{{\left(x^2+2\right)}^2\frac{d}{dx}2x-2x\frac{d}{dx}{\left(x^2+2\right)}^2}{{\left(x^2+2\right)}^4}\right]$

$=\frac{-\left[2{\left(x^2+2\right)}^2-2x\cdot 2\left(x^2+2\right)\cdot 2x\right]}{{\left(x^2+2\right)}^4}.$

$=\frac{-\left[2{\left(x^2+2\right)}^2\left[x^2+2-4x^2\right]\right]}{{\left(x^2+2\right)}^4}$

$=\frac{-2\left(x^2+2\right)\left(2-3x^2\right)}{{\left(x^2+2\right)}^4}=-\frac{2\left(2-3x^2\right)}{{\left(x^2+2\right)}^3}.$

At extreme points, g'(x) = 0.

$\Rightarrow \frac{-2x}{{\left(x^2+2\right)}^2}=0.$

$\Rightarrow$ x = 0.

At, x = 0, $g^{\prime \prime }(x)=\frac{-2\left(2-2\times 0^2\right)}{{\left(0^2+2\right)}^3}=\frac{-2(2-0)}{2^3}=-\frac{1}{2}<0$

∴x = 0 is a point of local maxima and value of local maximum is given by $g(0)=\frac{1}{0^2+2}=\frac{1}{2}.$

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<p><strong>(i) </strong>we have, f(x) = <em>x</em>².</p><p>f(x) = 2x.</p><p>andf(x) = 2.</p><p>AR extreme point, f(x) = 0</p><p>2x = 0</p><p>x = 0.</p><p>When x = 0, f(0) = 2 &gt; 0.</p><p>∴x = 0 is a point of local minima and value of local minimum is given by f(0) = 0² = 0.</p><p><strong>(ii)</strong> g(x) = <em>x</em>³ 3x</p><p>A(ii)</p><p>we have, g(x) = <em>x</em>³- 3<em>x</em></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> g'(x) = 3<em>x</em>²- 3</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> g''(x) = 6x.</p><p>At extreme point,</p><p>g'(x) = 0</p><p>3<em>x</em>²- 3 = 0.</p><p>3(<em>x</em>²- 1) = 0 ⇒ 3(x - 1)(x + 1) = 0.</p><p>x = 1 or x = -1.</p><p>At x = 1, g"(1) = 6.1 = 6 &gt; 0.</p><p>So, x = 1 is a point of local minima and value of local minimum is given by g(1) = 1³- 3.1 = 1 - 3 = - 2.</p><p>And at x = -1, g"( -1) = 6 ( -1) = 6 &lt; 0.</p><p>So, x = -1 is a point of local minima and value of local minimum is given by</p><p>g(- 1) = (- 1)³- 3(- 1) = 1 + 3 = 2.</p><p><strong>(iii)</strong>&nbsp;<span title="Click to copy mathml"><math><mrow><mi>h</mi><mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi><mo>+</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi><mo>.</mo><mn>0</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span></p><p>A(iii)</p><p>we have, h(x) = sin x + cos x.</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> h'(x) = cos x - sin x</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> h'(x) = -sin x - cos x.</p><p>At extreme points,</p><p>h'(x) = 0</p><p>cosx - sin x = 0</p><p>cosx = sin x</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mtext> </mtext><mfrac><mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi></mrow><mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi></mrow><mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span>&nbsp;1 = tan x</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mi>t</mi><mi>a</mi><mi>n</mi><mi>x</mi><mo>=</mo><mi>t</mi><mi>a</mi><mi>n</mi><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>.</mo></mrow></math></span></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1734068563phpiAlP6W_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1734068563phpiAlP6W.jpeg" alt="" width="502" height="132"></picture></div></div><p><strong>(iv)&nbsp;</strong><span title="Click to copy mathml"><math><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi><mo>,</mo><mn>0</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>2</mn><mi>π</mi></mrow></math></span></p><p>A(iv)</p><p>we have, f(x) = sin x cos x.</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span>&nbsp;f(x) = cos x + sin x</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span>&nbsp;f(x) = sin x + cos x.</p><p>At extreme points,</p><p>f(x) = 0</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span>&nbsp;cosx + sin x = 0.</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mfrac><mrow><mi>s</mi><mi>i</mi><mi>n</mi><mi>x</mi></mrow><mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi></mrow><mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>x</mi></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mi>t</mi><mi>a</mi><mi>n</mi><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn><mo>=</mo><mo>−</mo><mi>t</mi><mi>a</mi><mi>n</mi><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>=</mo><mi>t</mi><mi>a</mi><mi>n</mi><mrow><mo>(</mo><mrow><mi>π</mi><mo>−</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mi>t</mi><mi>a</mi><mi>n</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>π</mi><mo>−</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>∴</mo><mi>x</mi><mo>=</mo><mrow><mo>(</mo><mrow><mi>π</mi><mo>−</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>&nbsp;IInd quadrate or&nbsp;<span title="Click to copy mathml"><math><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>π</mi><mo>−</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>&nbsp;IIIth quadrate</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi>π</mi><mo>−</mo><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>8</mn><mi>x</mi><mo>−</mo><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac><mrow><mi>f</mi><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>.</mo></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mi>A</mi><mi>t</mi><mtext> </mtext><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mrow><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>s</mi><mi>i</mi><mi>n</mi><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mi>c</mi><mi>o</mi><mi>s</mi><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>=</mo><mo>−</mo><mi>s</mi><mi>i</mi><mi>n</mi><mrow><mo>(</mo><mrow><mi>π</mi><mo>−</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mi>o</mi><mi>s</mi><mrow><mo>(</mo><mrow><mi>π</mi><mo>−</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>=</mo><mo>−</mo><mi>s</mi><mi>i</mi><mi>n</mi><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>−</mo><mi>c</mi><mi>o</mi><mi>t</mi><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span><span title="Click to copy mathml"><br></span></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1734068728phpwMIpcn_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1734068728phpwMIpcn.jpeg" alt="" width="426" height="371"></picture></div><div><strong>(v)</strong> f(x) = <em>x</em>³-&nbsp;6<em>x</em>² + 9<em>x</em> + 15.</div></div><p>A(v)</p><p>we have, f(x) = <em>x</em>³- 6<em>x</em>² + 9<em>x</em> + 15.</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> f(x) = 3<em>x</em>²- 12<em>x</em> + 9.</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> f'(x) = 6x - 12.</p><p>At extreme point, f'(x) = 0.</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> 3<em>x</em>²- 12<em>x</em> + 9 = 0.</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> <em>x</em>²- 4<em>x</em> + 3 = 0</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> <em>x</em>²-<em>x </em>- 3<em>x</em> + 3 = 0</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> x(x - 1) -3(x - 1) = 0</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> (x - 1)(x - 3) = 0</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span>&nbsp;x = 1 or x = 3.</p><p>At, x = 1,f"(1) = 6 × 1 - 12 = 6 - 12 = - 6 &lt; 0</p><p>∴x = 1 is a point of local maxima and the value of local</p><p>Maximum is given by f(1) = 1³- 6(1)² + 9(1) + 15</p><p>= 1 - 6 + 9 + 15</p><p>= 19.</p><p>And at x = 3,f"(3) = 6 × 3 - 12 = 18 - 12 = 6 &gt; 0</p><p>∴x = 3 is a point of local minima and the value of</p><p>local minimum is given by f (3) = 3³- 6(3)² + 9(3) + 15.</p><p>= 27 - 54 + 27 + 15.</p><p>= 15.</p><p><strong>(vi)</strong>&nbsp;<span title="Click to copy mathml"><math><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>x</mi></mrow></mfrac><mo>,</mo><mi>x</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span></p><p>A(vi)</p><p>We have,&nbsp;<span title="Click to copy mathml"><math><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>x</mi></mrow></mfrac><mo>,</mo><mi>x</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mtext> </mtext><msup><mrow><mi>g</mi></mrow><mrow><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mi>g</mi><msup><mrow><mi>g</mi></mrow><mrow><mo>′</mo><mi>z</mi></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>4</mn></mrow><mrow><mn>2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><msup><mrow><mi>g</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>.</mo></mrow></math></span></p><p>At, extreme point, g'(x) = 0</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span> x = -2 or x = 2</p><p>Given, that x &gt; 0, hence we have x = 2.</p><p>At, x = 2,&nbsp;<span title="Click to copy mathml"><math><mrow><msup><mrow><mover accent="true"><mi>g</mi><mo>?</mo></mover></mrow><mrow><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>&gt;</mo><mn>0</mn></mrow></math></span></p><p>∴x = 2 is a point of local minima and value of local</p><p>Minimum is given by g(2) =&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>=</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>=</mo><mn>2</mn><mo>.</mo></mrow></math></span></p><p><strong>(vii)</strong>&nbsp;<span title="Click to copy mathml"><math><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math></span></p><p>A(vii)</p><p>Given,&nbsp;<span title="Click to copy mathml"><math><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>.</mo></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mrow><mo>[</mo><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mn>2</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>x</mi><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac></mrow><mo>]</mo></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mo>−</mo><mrow><mo>[</mo><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>⋅</mo><mn>2</mn><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>2</mn><mi>x</mi></mrow><mo>]</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac><mo>.</mo></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mo>−</mo><mrow><mo>[</mo><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>[</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mo>−</mo><mn>4</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>.</mo></mrow></math></span></p><p>At extreme points, g'(x) = 0.</p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo></mrow></math></span>&nbsp;x = 0.</p><p>At, x = 0,&nbsp;<span title="Click to copy mathml"><math><mrow><msup><mrow><mi>g</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>2</mn><mo>×</mo><msup><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>2</mn><mo stretchy="false">(</mo><mn>2</mn><mo>−</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>&lt;</mo><mn>0</mn></mrow></math></span></p><p>∴x = 0 is a point of local maxima and value of local maximum is given by&nbsp;<span title="Click to copy mathml"><math><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>.</mo></mrow></math></span></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1734069203phpSS4at0_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1734069203phpSS4at0.jpeg" alt="" width="319" height="295"></picture></div></div><div><div><picture><img src="https://images.shiksha.com/mediadata/images/articles/1734069223phptw07x9.jpeg" alt="" width="389" height="462"></picture></div></div>

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