Number of critical points How do you know if a function has infinite critical points?
I have learned that you can find critical points by setting the derivative of separate axes to 0 and solving, but I also think a function like sine would have an infinite amount of critical points whereas a parabola would only have 1, regardless of domain.
 A: A critical point is a point where the derivative of a function is either zero (there is going to be a horizontal tangent line to the curve at that point) or does not exist (it's not possible to draw a tangent line at that point). Take the example offered by saulspatz in the comments section. If you want to find all critical points that the sine function has, all you need to do is take its derivative and find all points where the derivative is either zero or does not exist.
$$
\left(\sin{x}\right)'=\cos{x}\\
$$
Now, set the derivative equal to zero:
$$\cos{x}=0$$
All $x$'s that solve this equation plus all $x$'s where the function $\cos{x}$ is not defined are going to be the critical points of $\sin{x}$. $\cos{x}$ is defined everywhere. So, there are not points where the derivative of $\sin{x}$ does not exit. Solving $\cos{x}=0$ for $x$ gives us this infinite solution set: $\frac{\pi}{2}+\pi k$ where $k\in Z$. So, the sine function has an infinite number of critical points. And that's how you know if a function has an infinite number of critical points. You literally need to go and figure that out.
A: If a function has a derivative that has infinite roots then the original function will have infinite special points. But, if a function has a derivative with finite roots, for example a polynomial, then the function will have finite special points.
