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.

HINT: we have for 1) $$f'(x)=4x^3\sin\left(\frac{1}{x}\right)+x^4\cos\left(\frac{1}{x}\right)\cdot \left(-\frac{1}{x^2}\right)$$
and for 2)$$g'(x)=4x^3\left(2+\sin\left(\frac{1}{x}\right)\right)+x^4\cdot\cos\left(\frac{1}{x}\right)\left(-\frac{1}{x^2}\right)$$
and 3)$$h'(x)=4x^3\left(-2+\sin\left(\frac{1}{x}\right)\right)+x^4\cos\left(\frac{1}{x}\right)\left(-\frac{1}{x^2}\right)$$
additionally use that $$|f(x)|\le x^4$$
and for $$f(x)$$ we have a sign Change from minus to plus
$g(x)$ has a sign change from plus to plus
$h(x)$ has a sign change for minus to minus.....
