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

JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Continuity and Differentiability

Let

m

and

n

be the number of points at which the function

f (x) = max {x, x^{3}, x^{5}, \dots, x^{21}}

x \in R

, is not differentiable and not continuous, respectively. Then

m + n

is equal to ______.

Answer: 3

Among the odd powers

x, x^{3}, \dots, x^{21}

, the largest is

x

for

x < - 1

x^{21}

for

- 1 \leq x < 0

x

for

0 \leq x < 1

, and

x^{21}

for

x \geq 1

. This maximum is continuous everywhere, so there are no points of discontinuity. At

x = - 1

the one-sided derivatives are

1

and

21

; at

x = 0

they are

0

and

1

; at

x = 1

they are

1

and

21

; so the function is not differentiable at three points. With

m = 3

and

n = 0

, the value of

m + n

3

JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Integer answer