Find the critical numbers of $f(x) = 7x^3 + |x|$. Determine critical numbers at which the tangent line is horizontal
Here is what I have so far:
I know that $|x|$ can be rewritten as $\sqrt{x^2}$
Differentiating $f(x) = 7x^3 + |x| $
where $f^\prime(x) = 21x^2 + \frac{x}{|x|} $
I have know that a critical number is when $f^\prime(x) = 0$ or undefined.
The first critical number would be 0 since it would be undefined. The second critical number would be $\frac{-1}{\sqrt{21}}$. This is also supported when I had graphed the derivative so the tangent line should be horizontal at those two critical numbers. Unfortunately, I'm not even sure if I found all of the critical numbers. Sorry for the format.