Question

Let $f(x)$ be a non-constant differentiable function satisfying $f(x)=x^2-\underset{0}{\overset{1}{\int }}(f(t)+x{)}^{2}dt$. Then which of the following is/are correct?

• A. $f(x)$ is monotonic.
• B. $\underset{x\to \frac{4}{3}}{\mathrm{Lim}}(f(x){)}^{\mathrm{cosec}(3\pi x)}$ 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)

$f(x)=x^2-\underset{0}{\overset{1}{\int }}{f}^2(t)dt-2x\underset{0}{\overset{1}{\int }}f(t)dt-x^2\underset{0}{\overset{1}{\int }}dt$
$f(x)=x^2+A+2Bx-x^2=A+2Bx$
$A=-\underset{0}{\overset{1}{\int }}{f}^2(t)dt=-\underset{0}{\overset{1}{\int }}(A+2Bt{)}^{2}dt$
$B=-\underset{0}{\overset{1}{\int }}f(t)dt=-\underset{0}{\overset{1}{\int }}(A+2Bt)dt$
$\therefore f(x)=3x-3$
Now verify all the option.