QJEEM19Ma9068: JEE Main 2019 Mathematics - Class 12 JEEM19Ma9068 Solution: Let f be a differentiable function from R to R such that…

Question

Verified Solution • 2025–26 Session moderate

Let f be a differentiable function from

R to R such that

\left| {f\left( x \right) - f\left( y \right)} \right| \le 2{\left| {x - y} \right|^{{3 \over 2}}},

for all

x,y \in

R.

If

f\left( 0 \right) = 1

then

\int\limits_0^1 {{f^2}} \left( x \right)dx

is equal to :

Accepted Answer

\left| {f(x) - f(y)} \right| \le 2{\left[ {x - y} \right]^{3/2}}

\left| {{{f(x) - f(y)} \over {x - y}}} \right| \le 2{\left| {x - y} \right|^{1/2}}

\mathop {\lim }\limits_{y \to x} \left| {{{f(x) - f(y)} \over {x - y}}} \right| \le \mathop {\lim }\limits_{y \to x} 2{\left| {x - y} \right|^{1/2}}

\Rightarrow \left| {f'\left( x \right)} \right| \le 0

\Rightarrow f'\left( x \right) = 0

\Rightarrow f\left( x \right) =

constant

as

f\left( 0 \right) = 1 \Rightarrow f\left( x \right) = 1

\int\limits_0^1 {{f^2}} \left( x \right)dx = 1

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