First slide

Monotonicity

Question

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

Difficult

A

$f^{'} (0) = - \frac{π}{2}$
B

$f is not differentiable at x = 0$
C

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

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

Solution

$\begin{array}{l} f (x) = x (π - \cos^{- 1} (\sin | x |)) = x (π - (\frac{π}{2} - \sin^{- 1} (\sin | x |))\} = x (\frac{π}{2} + | x |) \\ f (x) = \{\begin{array}{l} x (\frac{π}{2} + x) & x \geq 0 \\ x (\frac{π}{2} - x) & x < 0 \end{array} \Rightarrow f^{'} (x) = \{\begin{cases} (\frac{π}{2} + 2 x) & x \geq 0 \\ (\frac{π}{2} - 2 x) & x < 0 \end{cases} \\ f^{'} (x) is increasing in (0, \frac{π}{2}) and decreasing in (\frac{- π}{2}, 0) \end{array}$

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