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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.

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  • $\begingroup$ Use $|x|=x$ when $x\gt 0$ and $|x|=-x$ when $x\lt 0$ and do the calculations separately. $\endgroup$ Oct 30, 2015 at 3:17
  • $\begingroup$ That was my first approach at this problem. So $f(x) = 7x^3 + x$ and $f(x)= 7x^3 - x$ then, $f'(x) = 21x^2 + 1$ and $f'(x) = 21x^2 - 1$ However, setting the first one would result in a negative in square root. The second one would be 1/sqrt(21) $\endgroup$
    – LRo
    Oct 30, 2015 at 3:21
  • $\begingroup$ You found all the critical numbers. Beside the origin, there is only $-1/\sqrt{21}$. $\endgroup$ Oct 30, 2015 at 3:24
  • $\begingroup$ Thanks everyone for the help! I guess I just needed some clarifications. $\endgroup$
    – LRo
    Oct 30, 2015 at 3:31

2 Answers 2

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$f'$ isn't defined at $x=0$, so there's one critical point. Now for $x\neq 0$, $\frac{x}{|x|}=1$ or $-1$. In the first case, you're find roots of $21x^2+1$. In the second case, you're finding roots of $21x^2-1$.

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$$f^\prime(x) = 21x^2 + \frac{x}{|x|} = 0 \quad \implies \frac{x}{|x|}= -21x^2 $$

At $x=0$ the derivative is undefined. That's one critical point.

For $x>0$, the latter equation becomes $1= -21x^2$. This has no real solutions.

For $x<0$, the equation becomes $-1= -21x^2$. That gives $$x = \pm\frac{1}{\sqrt{21}}$$

Check to see if these both work... and only one does. It looks like you have all the points, given that there are no endpoints to consider.

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