Question

Let a function $f\left(x\right)$ satisfies $f^2\left(x\right)-f^2\left(y\right)=4(x-y)$ and $f\left(0\right)=2(f\left(x\right)\ge 0)$whose domain is $[a,\infty )$ and it is differentiable on $(a,\infty )$

The value of definite integral ${\int }_a^{0}f\left(x\right)dx$ is

• A. $\frac{3}{4}$
• B. $\frac{4}{3}$
• C. $\frac{4}{5}$
• D. $\frac{5}{4}$

Answer 1

The correct option is B $\frac{4}{3}$

Let $f\left(x\right)=b\sqrt{x+a}$

Then $f^2\left(x\right)-f^2\left(y\right)=b^2(x-y)=4(x-y)$

hence $b^2=4$
$b=2$ ...(i)

And $f\left(0\right)=2$

Then $b\sqrt{a}=2$
$2\sqrt{a}=2$
$a=1$.

Hence $f\left(x\right)=2\sqrt{x+1}$

Now $x+1\ge 0$
$xϵ[-1,\infty )$.
Hence ${\int }_{-1}^{0}2\sqrt{x+1}$
$=2\frac{2}{3}\left[\right(x+1{)}^{\frac{3}{2}}{]}_{-1}^0$
$=\frac{4}{3}$.