Showing the local maximum or minimum, while the function changes sign infinitely often Please I need a hand in solving this problem:
These 3 functions' values at $0$ are all $0$ and for $x\ne0$, $$f(x)=x^4\sin\frac{1}{x}, \, g(x)=x^4\left(2+\sin\frac{1}{x}\right), \, h(x)=x^4\left(-2+\sin\frac{1}{x}\right)$$
b- Show that $f$ has neither a local maximum nor a local minimum
at $0$, $g$ has a local minimum, and $h$ has a local
maximum.
The derivatives of these functions change sign infinitely often on both sides of $0$. I couldn't use the 1st nor the 2nd derivative test.

The behavior of these functions is easily handled without using derivatives. Note that for any given open interval $I$ containing $0$ we can find positive integer $ n$ such that points $c, d$ given by $$c=\frac{2}{(4n+1)\pi},d=\frac{2}{(4n+3)\pi}$$ lie in $I$ and $f(c) >0=f(0)>f(d)$. So $0$ is neither a local maximum nor a local minimum of $f$.
Functions $g, h$ are easier to handle. For $g$ note that $(2+\sin(1/x))>0$ so that $g(x)$ is positive for all $x$ except $x=0$ and similarly $h(x) $ is always negative except at $x=0$. The desired conclusion follows immediately. 
