$$h(x)=\begin{cases}
{4x+2x^2 \sin \left(\frac{1}{x}\right)}{} & \text{if $x <0$} \\[6pt]
{x}\cdot{\cos(x)}+ kx & \text{if $x\geq0$} \\
\end{cases}
$$
I know how to prove they are continuous but how do I show the function is differentiable? Any ideas please? I think $k=3$
Best Answer
Let $f$ be such a function, you may try to expand the details about \begin{align*} \lim_{h\rightarrow 0^{+}}\dfrac{f(h)-f(0)}{h}=\lim_{h\rightarrow 0^{-}}\dfrac{f(h)-f(0)}{h} \end{align*} and solve for $k$.