Prove that the function f given by f ( x ) = x is strictly increasing on ( 0, 2 ) and strictly decreasing on ( 2 , ) .

We have, f(x) = ( x) …(i) Differentiating (i) w.r.t. x , we get f'(x) = x ( x) = x As x > 0 for all x ( 0, 2 ) and x < 0 for all x ( 2 , ) Therefore, f(x) is strictly increasing on ( 0, 2 ) and strictly decreasing on ( 2 , ) .

Application of Derivatives — Class 12 Maths Solution

ncert exercise SA NCERT Ex. 6.2, Q.16,Page 206

Question

Prove that the function $f$ given by $f\left( x \right) = \log \sin x$ is strictly increasing on $\left( {0,\cfrac{\pi }{2}} \right)$ and strictly decreasing on $\left( {\cfrac{\pi }{2},\pi } \right)$.

Step-by-step Solution

We have, $f(x) = \log (\sin x)$ …(i)

Differentiating (i) w.r.t. $x$, we get $f'(x) = \cfrac{1}{{\sin x}}(\cos x) = \cot x$

As $\cot x > 0$ for all $x \in \left( {0,\cfrac{\pi }{2}} \right)$ and $\cot x < 0$ for all $x \in \left( {\cfrac{\pi }{2},\pi } \right)$

Therefore, $f(x)$ is strictly increasing on $\left( {0,\cfrac{\pi }{2}} \right)$ and strictly decreasing on $\left( {\cfrac{\pi }{2},\pi } \right)$.

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