Suppose a differentiable function f(x) satisfies the identity f(x+y)=f(x)+f(y)+xy^2+x^2y f…

Question

Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y)=f(x)+f(y)+xy^2+x^2y$ for all real $x,y$. If $\displaystyle\lim_{x	o0}\dfrac{f(x)}{x}=1$, then $f'(3)$ is equal to \_\_\_\_\_\_\_.

Answer: 10

Accepted Answer

Differentiating w.r.t. $x$: $f'(x+y)=f'(x)+y^2+2xy$. Setting $y=-x$: $f'(0)=f'(x)+x^2-2x^2=f'(x)-x^2$.

Since $\displaystyle\lim_{x	o0}\dfrac{f(x)}{x}=1$, $f(0)=0$ and $f'(0)=1$.

So $f'(x)=1+x^2$, and $f'(3)=1+9=10$.

Solutions · 1