Let <math> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>x</mi> <msup> <mrow> <mrow> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> <mo>?</mo> <mo>(</mo> <mo>-</mo> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mo>?</mo> <mo>|</mo> <mi>x</mi> <mo>|</mo> <mo>)</mo> <mo>,</mo> <mi>x</mi> <mo>∈</mo> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <mo>-</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>,</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </math> <math><msup><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mrow><mfrac><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></mfrac><mfrac><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></mfrac></mrow></mrow></mfenced></math> , then which of the following is true?

f ' ( x ) = x π - c o s - 1 ? ( s i n ? | x | ) = x π - π 2 - s i n - 1 ? ( s i n ? | x | ) = x π 2 + | x | f ( x ) = x π 2 + x x ≥ 0 x π 2 - x x < 0 f ' ( x ) = π 2 + 2 x x ≥ 0 π 2 - 2 x x < 0 is increasing in f ' ( x ) and decreasing in 0 , π 2

Let <math> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>x</mi> <msup> <mrow> <mrow> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> <mo>?</mo> <mo>(</mo> <mo>-</mo> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mo>?</mo> <mo>|</mo> <mi>x</mi> <mo>|</mo> <mo>)</mo> <mo>,</mo> <mi>x</mi> <mo>∈</mo> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <mo>-</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>,</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </math> <math><msup><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mrow><mfrac><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></mfrac><mfrac><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></mfrac></mrow></mrow></mfenced></math>&nbsp;, then which of the following is true?

<math> <mi>f</mi> </math> is not differentiable at <math> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math>

<math> <mi>f</mi> </math> is decreasing in <math> <mfenced separators="|"> <mrow> <mrow> <mo>-</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>,</mo> <mn>0</mn> </mrow> </mrow> </mfenced> </math> and increasing in <math> <mfenced separators="|"> <mrow> <mrow> <mn>0</mn> <mo>,</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </math>

<math> <mi>f</mi> </math> is increasing in <math> <mfenced separators="|"> <mrow> <mrow> <mo>-</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>,</mo> <mn>0</mn> </mrow> </mrow> </mfenced> </math> and decreasing in <math> <mfenced separators="|"> <mrow> <mrow> <mn>0</mn> <mo>,</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </math>

Ask & Answer Question

Chemistry Class 12th Ncert Solutions Chemistry Class 12th

Let $f (x) = x {c o s}^{- 1} ? (- s i n ? | x |), x \in [- \frac{π}{2}, \frac{π}{2}]$ $^{} [\frac{}{} \frac{}{}]$ , then which of the following is true?

Option 1 -

$f^{'} (0) = - \frac{π}{2}$

Option 2 -

$f$ is not differentiable at $x = 0$

Option 3 -

$f$ is decreasing in $(- \frac{π}{2}, 0)$ and increasing in $(0, \frac{π}{2})$

Option 4 -

$f$ is increasing in $(- \frac{π}{2}, 0)$ and decreasing in $(0, \frac{π}{2})$

3 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

Raj Pandey | Contributor-Level 9

2 months ago

Correct Option - 2

Detailed Solution:

$f^{'} (x) = x (π - {c o s}^{- 1} ? (s i n ? | x |)) = x (π - (\frac{π}{2} - {s i n}^{- 1} ? (s i n ? | x |))) = x (\frac{π}{2} + | x |)$

$f (x) = \{\begin{array}{l} x (\frac{π}{2} + x) & x \geq 0 \\ x (\frac{π}{2} - x) & x < 0 \end{array}$

$f^{'} (x) = \{\begin{array}{l} \frac{π}{2} + 2 x & x \geq 0 \\ \frac{π}{2} - 2 x & x < 0 \end{array}$ is increasing in $f^{'} (x)$ $^{}$ and decreasing in $(0, \frac{π}{2})$

<math> <mrow> <mi mathvariant="normal"> </mi> </mrow> <msup> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">'</mi> </mrow> </mrow> </msup> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>x</mi> <mfenced separators="|"> <mrow> <mrow> <mi>π</mi> <mo>-</mo> <msup> <mrow> <mrow> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> <mo>?</mo> <mo>(</mo> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mo>?</mo> <mo>|</mo> <mi>x</mi> <mo>|</mo> <mo>)</mo> </mrow> </mrow> </mfenced> <mo>=</mo> <mi>x</mi> <mfenced separators="|"> <mrow> <mrow> <mi>π</mi> <mo>-</mo> <mfenced separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>-</mo> <msup> <mrow> <mrow> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> <mo>?</mo> <mo>(</mo> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mo>?</mo> <mo>|</mo> <mi>x</mi> <mo>|</mo> <mo>)</mo> </mrow> </mrow> </mfenced> </mrow> </mrow> </mfenced> <mo>=</mo> <mi>x</mi> <mfenced separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>+</mo> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> </mfenced> </math> <math> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mfenced open="{" close="" separators="|"> <mrow> <mrow> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mi>x</mi> <mfenced separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>x</mi> </mrow> </mrow> </mfenced> </mrow> </mtd> <mtd> <mrow> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>x</mi> <mfenced separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>-</mo> <mi>x</mi> </mrow> </mrow> </mfenced> </mrow> </mtd> <mtd> <mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> </mfenced> </math> <math> <msup> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">'</mi> </mrow> </mrow> </msup> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mfenced open="{" close="" separators="|"> <mrow> <mrow> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mi>x</mi> </mrow> </mtd> <mtd> <mrow> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>-</mo> <mn>2</mn> <mi>x</mi> </mrow> </mtd> <mtd> <mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> </mfenced> </math>  is increasing in <math> <msup> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">'</mi> </mrow> </mrow> </msup> <mo>(</mo> <mi>x</mi> <mo>)</mo> </math> <math><msup><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msup></math> and decreasing in <math> <mfenced separators="|"> <mrow> <mrow> <mn>0</mn> <mo>,</mo> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </math>

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Let $f (x) = x {c o s}^{- 1} ? (- s i n ? | x |), x \in [- \frac{π}{2}, \frac{π}{2}]$ $^{} [\frac{}{} \frac{}{}]$ , then which of the following is true?

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Let f ( x ) = x c o s - 1 ? ( - s i n ? | x | ) , x ∈ - π 2 , π 2 , then which of the following is true?

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Let $f (x) = x {c o s}^{- 1} ? (- s i n ? | x |), x \in [- \frac{π}{2}, \frac{π}{2}]$ $^{} [\frac{}{} \frac{}{}]$ , then which of the following is true?