Prove that $f(x) =∇f(0)·x$

Question

For f - differentiable function, with the property $ f(tx)=t f(x)$ for all real t and all $x$ in $\mathbb R^n$, prove that $f(x) =∇f(0)·x$, where $∇f$ is the gradient of $f$.

Answer 1

Let $x$ be fixed .

Then we have $\frac{d}{dt}f(tx)= \frac{d}{dt}(tf(x))$.

Compute these derivatives and then plug in $t=0.$