Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x) - f(y)| \le 2|x - y{|^{3/2}}{\rm{ }}\,{\rm{for}}\,\,{\rm{ all}}\,{\rm{ }}x,y \in R{\rm{. }}$ If $f(0)=1$, then $\int_0^1 f^2(x) d x$ is equal to

JEE MATHEMATICS MCQ: DIFFERENTIATION

1 Medium

1 of 1

Analyzing +4 | -1

Let

f

be a differentiable function from

R

R

such that

|f(x) - f(y)| \le 2|x - y{|^{3/2}}\mathrm{ }\,\mathrm{for}\,\,\mathrm{ all}\,\mathrm{ }x,y \in R\mathrm{. }

f(0)=1

, then

\int_0^1 f^2(x) d x

is equal to

View answer and explanation

Correct option: D

Option D (1) is correct because the given condition implies that the derivative of

f(x)

is zero everywhere, making

f(x)

a constant function. Since

f(0)=1

f(x)=1

for all

x

, and the integral of

f^2(x)

from 0 to 1 is 1.