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 $f^{{}^{\prime }}\left(x\right)$ is equals to

• A. 9
• B. 3
• C. 27
• D. 6

Answer 1

The correct option is C 27

$f^{{}^{\prime }}x=\underset{h\to 0}{Lt}\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{Lt}\frac{f\left(x\right)f\left(h\right)-f\left(x\right)}{h}=\underset{h\to 0}{Lt}\frac{(2h^2+3h)g\left(h\right)}{h}=\underset{h\to 0}{Lt}(2h+3)g\left(h\right)=3g\left(0\right)=3.9=27$