Let $\displaystyle f:\mathbb{R}\rightarrow\mathbb{R}$ be defined by $\displaystyle f(x) = 2x^4+x^4\sin{1/x}$ for $\displaystyle x\neq0$ and $\displaystyle f(0)=0$ .

Show that f has an absolute minimum at $\displaystyle x=0$, but that its derivative has both positive and negative values in every neighborhood of 0 .