Let f : R \rightarrow R be defined as f (x) = | x | + | x2 – 1 |. The total number of points at which f attains either a local maximum or a local minimum is

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Maxima and Minima

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Let f : R $\to$ R be defined as f (x) = | x | + | x² – 1 |. The total number of points at which f attains either a local maximum or a local minimum is

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f : R $\to$ R

f (x) = | x | + | (x – 1) (x + 1) | = ${\begin{cases} - x + x^{2} - 1, x < - 1 \\ - x - x^{2} + 1, - 1 \leq x < 0 \\ x - x^{2} + 1, 0 \leq x < 1 \\ x + x^{2} - 1, x \geq 1 \end{cases}$

$f' (x) = {\begin{cases} 2 x - 1, x < - 1 \\ - 2 x - 1, - 1 < x < 0 \\ 1 - 2 x, 0 < x < 1 \\ 1 + 2 x, x > 1 \end{cases}$

$∴$ Total number of points = 5.

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