Question

Let $f\left(x\right)$ is a differentiable function such that $f\left(x + y\right)=f\left(x\right)+f\left(y\right)+2xy\forall x,y\in R$ and $\underset{x \rightarrow 0}{lim}\frac{f \left(x\right)}{x}=210$ , then $f\left(2\right)$ is equal to

• A. $20$
• B. $105$
• C. $424$
• D. none of these

Answer 1

Answer: (C)

$f^{'}\left(x\right)=\underset{h \rightarrow 0}{lim}\frac{f \left(x + h\right) - f \left(x\right)}{h}$
$=\underset{h \rightarrow 0}{lim}\frac{f \left(x\right) + f \left(h\right) + 2 x h - f \left(x\right)}{h}$
$f^{'}\left(x\right)=\underset{h \rightarrow 0}{lim}\left(\frac{f \left(h\right)}{h} + 2 x\right)=210+2x$
$f\left(x\right)=210x+x^{2}+C$
$f\left(0\right)=0\Rightarrow C=0$
$f\left(x\right)=210x+x^{2}$
$\therefore f\left(2\right)=424$