Question

Let $f(x)$ be a non-constant differentiable function satisfying $f(x)=x^2-\int\limits_0^1(f(t)+x)^2 d t$. Then which of the following is/are correct?

• A. $f(x)$ is monotonic.
• B. $\underset{x \rightarrow \frac{4}{3}}{\text{Lim}} (f(x))^{\operatorname{cosec}(3 \pi x)}$ is equal to $e^{\frac{1}{\pi}}$.
• C. Derivative of $f(x)$ w.r.t $\ln x$ at $x=1$ is equal to 3 .
• D. Area bounded by $f(x)$ and co-ordinate axes is $\frac{2}{3}$.

Answer 1

Answer: (A), (B), (C)

$f(x)=x^2-\int\limits_0^1 f^2(t) d t-2 x \int\limits_0^1 f(t) d t-x^2 \int\limits_0^1 d t$
$f(x)=x^2+A+2 B x-x^2=A+2 B x $
$A=-\int\limits_0^1 f^2(t) d t=-\int\limits_0^1(A+2 B t)^2 d t $
$B=-\int\limits_0^1 f(t) d t=-\int\limits_0^1(A+2 B t) d t $
$\therefore f(x)=3 x-3$
Now verify all the option.