Question

Let $f\left(x\right)$ be continous and differentiable function for all real values.
$f(x+y)=f\left(x\right)-3xy+f\left(y\right)$. If $\underset{h\to 0}{lim}\frac{f\left(h\right)}{h}=7$, then value of $f^{\prime }\left(x\right)$ is

• A. $-3x$
• B. $7$
• C. $-3x+7$
• D. $2f\left(x\right)+7$

Answer 1

Answer: (C)

$-3x+7$

$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lim}\frac{f\left(x\right)+f\left(h\right)-3hx-f\left(x\right)}{h}=\underset{h\to 0}{lim}\frac{f\left(h\right)}{h}-3x=-3x+7$