Question

Let f be a function such that f(x+y)=f(x)+f(y) for all x and y and $f\left(x\right)=(2x^2+3x)g\left(x\right)$ for all x where g(x) is continuous and g(0)=9 then f'(0) is equals to

• A. 9
• B. 3
• C. 27
• D. 6

Answer 1

The correct option is C 27

$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
$=\underset{h\to 0}{lim}\frac{f\left(x\right)+f\left(h\right)-f\left(x\right)}{h}$
$=\underset{h\to 0}{lim}\frac{(2h^2+3h)g\left(h\right)}{h}$
$=\underset{h\to 0}{lim}(2h+3)g\left(h\right)$
$=3g\left(0\right)$
$=27$