Question

If $\underset{x\to a}{lim}\frac{x.{\int }_a^{x}f\left(x\right)dx}{x-a}=a^3+a^2$ $(a\ne 0)$ then $f\left(2a\right)$ is

• A. $a(2a+1)$
• B. $2a(2a+1)$
• C. $4a^2+1$
• D. $2a(a+1)$

Answer 1

The correct option is B $2a(2a+1)$

$\underset{x\to a}{lim}\frac{x.{\int }_a^{x}f\left(x\right)dx}{x-a}=a^3+a^2$
$f\left(a\right)\underset{x\to a}{lim}\frac{x.{\int }_a^{x}dx}{x-a}=a^3+a^2$
and
${\int }_a^{x}dx=x-a$
hence
$af\left(a\right)=a^3+a^2$
$\Rightarrow f\left(a\right)=a^2+a$
$f\left(2a\right)=4a^2+2a=2a(2a+1)$