How many extreme points of function $f: \mathbb{R} \mapsto \mathbb{R}, f(x) = |x^4 - 4x^3|$?
I have looked at materials about how to evaluate the extreme points of a function, but unfortunately, there are no literatures I find that explain about the function of absolute value like this problem. I tried to draw the graph by using app, and it looks like there are 2 extreme points, one is $x =0$. But, how can we find it precisely without looking at the graph? Thanks for your help
We know that $|y-a|=\begin{cases}y-a\qquad y\geq a\\-(y-a)\qquad y<a\end{cases}$. So we just have to apply this to the function $f$ as such: $$f(x)=|x^4-4x^3|=|x|^3|x-4|=\begin{cases}x^3(x-4)\qquad x<0\\-x^3(x-4)\qquad 0\leq x< 4\\x^3(x-4)\qquad x\geq 4\end{cases}$$ Note: the first case ($x<0$) gives $|x|=-x$ and $|x-4|=-(x-4)$, so the two negatives cancel out. The second case ($0\leq x<4$) gives $|x|=x$ and $|x-4|=-(x-4)$, so we get a negative sign. And the third case ($x\geq 4$) gives $|x|=x$ and $|x-4|=x-4$, so no negative signs.