Let f : (−∞, −1] → (a, b] be defined as $f\left(x\right)=e^{x^3-3x+1}$ , if f is both one and onto, then the distance from a point P(2a + 4, b + 2) to curve x + ye⁻³ − 4 = 0 is

Correct Option - 3


Detailed Solution:

  $f\left(x\right)=e^{x^3-3x+1}$

$f\text{'}\left(x\right)=e^{x^3-3x+1}$  (3x² − 3)

$e^{x^3-3x+1}$ = ⋅ 3(x −1)(x +1)

For x ∈ (−∞, −1], f '(x) ≥ 0

∴     f(x) is increasing function

∴     a = e^–∞ = 0 = f (−∞)

b = e⁻¹⁺³⁺¹ = e³ = f (−1)

∴     P(4, e³ + 2)

$d=\frac{\left(e^3+2\right)\left(e^{-3}\right)}{\sqrt[]{1+e^{-6}}}=\frac{1+2e^{-3}}{\sqrt[]{1+e^{-6}}}=\frac{1+2e^{-3}}{\sqrt[]{1+e^{-6}}}=\frac{e^3+2}{\sqrt[]{e^6+1}}$

<p><strong><span lang="EN-IN"> &lt;!-- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1820305028"> </o:OLEObject>&lt;/xml&gt;&lt;![endif]--&gt;&nbsp;<span contenteditable="false"> <math> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math> </span></span></strong></p><p><span lang="EN-IN"><span contenteditable="false"> <math> <mrow> <mi>f</mi> <mo>'</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math> </span> &nbsp;</span><span lang="EN-IN">(3x² − 3)</span></p><p><span lang="EN-IN"><span contenteditable="false"> <math> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math> </span>= <strong> &lt;!-- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1820305030"> </o:OLEObject>&lt;/xml&gt;&lt;![endif]--&gt;</strong></span><span lang="EN-IN">⋅ </span><span lang="EN-IN">3(x −1)(x +1)</span></p><p><span lang="EN-IN">For x&nbsp;</span><span lang="EN-IN">∈ </span><span lang="EN-IN">(−∞, −1], f '(x) ≥ 0</span></p><p><span lang="EN-IN">∴&nbsp;&nbsp;&nbsp;&nbsp; </span><span lang="EN-IN">f(x) is increasing function</span></p><p><span lang="EN-IN">∴&nbsp;&nbsp;&nbsp;&nbsp; </span><span lang="EN-IN">a = e^–∞ = 0 = f (−∞)</span></p><p><span lang="EN-IN">b = e⁻¹⁺³⁺¹ = e³ = f (−1)</span></p><p><span lang="EN-IN">∴&nbsp;&nbsp;&nbsp;&nbsp; </span><span lang="EN-IN">P(4, e³ + 2)</span></p><div><div><picture><source srcset="" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1758779837phpD2bpHn.jpeg" alt="" width="351" height="175" loading="lazy"></picture></div></div><p><span contenteditable="false"> <math> <mrow> <mi>d</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mroot> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mn>2</mn> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mroot> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mn>2</mn> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mroot> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mroot> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> </mrow> </math> </span></p>

0 Upvotes 0 Upvotes 0 Downvotes

• • Report Abuse
• Comment