Let m and n be number of points at which the function , is not differentiable and not continuous, respectively, Then m + n is equal to __________. [2025]

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Solution

Q.
Let m and n be number of points at which the function $f (x) = \max {x, x^{3}, x^{5}, . . ., x^{21}}, x \in R$ , is not differentiable and not continuous, respectively, Then m + n is equal to __________. [2025]

Ans.
(3)

Here, $f (x)$ $= {\begin{matrix} x, & x < - 1 \\ x^{21}, & - 1 \leq x < 0 \\ x, & 0 \leq x < 1 \\ x^{21}, & x \geq 1 \end{matrix}$ is continuous everywhere.

Then, n = 0

$\Rightarrow f' (x)$ $= {\begin{matrix} 1, & x < - 1 \\ 21 x^{20}, & - 1 \leq x < 0 \\ 1, & 0 \leq x < 1 \\ 21 x^{20}, & x \geq 1 \end{matrix}$ is not differentiable at

$x = - 1, 0, 1 \Rightarrow m = 3$

So, m + n = 3.

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