Question

f(x) = min{1,cos x, 1-sin x} , -π≤x≤π , then

A) f(x) is not differentiable at 0
B) f(x) is differentiable at π/2
C) f(x) has local maxima at 0
D) none of these​

Answer 1

we have f(x) = min{1,cos x, 1-sin x

therefore f(x) can be written as

$\displaystyle\sf \:\;\;\:\:\:\;\;\;\;f(x) = \begin{cases} \sf cos\:x,\;\;\;\;\bf -\dfrac{\boldsymbol\pi}{2}\leq x\leq 0\\\\\sf 1-sin\:x,\;\;\;\;\bf 0<x \leq \dfrac{\boldsymbol\pi}{2}\\\\\sf cos\:x,\;\;\;\;\bf \dfrac{\boldsymbol\pi}{2}< x < \boldsymbol\pi\end{cases}$

$\displaystyle\sf\implies f^{\prime}(x) = \begin{cases} \sf -sin\:x,\;\;\;\;\bf -\dfrac{\boldsymbol\pi}{2}\leq x\leq 0\\\\\sf -cos\:x,\;\;\;\;\bf 0<x \leq \dfrac{\boldsymbol\pi}{2}\\\\\sf-sin\:x,\;\;\;\;\bf \dfrac{\boldsymbol\pi}{2}< x < \boldsymbol\pi\end{cases}$ because f'(0) = 0

hence, f(x) has local maxima at 0 and is not differentiable at x = 0