Let $f : \to$ be defined as $f (x) = | x | + | x^{2} - 1 |$ . The total number of points at which f attains either a local maximum or a local minimum is

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Solution

$f (x) = | x | + | x^{2} - 1 |$

$f (x) = | x | + | (x - 1) (x + 1) |$

$f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} x^{2} - x - 1 & , & x < - 1 - x^{2} - x + 1 & , & - 1 \leq x < 0 - x^{2} + x + 1 & , & 0 \leq x < 1 x^{2} + x - 1 & , & x \geq 1 \end{matrix}$

$f^{'} (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 2 x - 1 & , & x < - 1 - 2 x - 1 & , & - 1 \leq x < 0 - 2 x + 1 & , & 0 \leq x < 1 2 x + 1 & , & x \geq 1 \end{matrix}$

$f (x)$ is not differentiable at $x = 0, \pm 1 and f^{'} (x) = 0 at x = - \frac{1}{2}, \frac{1}{2}$

Also sign scheme of $f^{'} (x)$

$x = - 1, 0, 1$ are point of minima

$x = \frac{- 1}{2}, \frac{1}{2}$ are point of maxima

total number of point of maxima
and minima = 5

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