Let f be a differentiable function from R to R such that |f(x) - f(y)| ≤ 2|x - y|^3/2, for all x, y ∈ R. If f(0) = 1, then int_0^1 f^2(x) dx is equal to

Let f be a differentiable function from R to R such that… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 9 Jan · Shift II

Let

f

be a differentiable function from

R

R

such that

|f(x) - f(y)| \leq 2|x - y|^{\frac{3}{2}}

, for all

x, y \in R

. If

f(0) = 1

, then

\int_{0}^{1} f^2(x)\, dx

is equal to

A.
2
B.
$\frac{1}{2}$
C.
1
D.
0

Show answer and solution

Correct option: C

Solution

From the condition, |f(x) - f(y)| ≤ 2|x - y|^(3/2). Dividing both sides by |x - y| and taking the limit as y → x gives |f'(x)| ≤ 2|x - y|^(1/2) → 0, so f'(x) = 0 for all x. Thus f is a constant function, and since f(0) = 1, f(x) = 1 everywhere. Therefore ∫₀¹ f²(x) dx = ∫₀¹ 1 dx = 1.

[JEE Main 2019 · 9 Jan · Shift II]

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