1. Maxima and Minima

Find all of the critical numbers of the function f(x)=|x^2+4x| on the open interval (-3,3).

2. Originally Posted by Velvet Love
Find all of the critical numbers of the function f(x)=|x^2+4x| on the open interval (-3,3).

$\displaystyle \frac{d}{dx} |u| = \frac{u}{|u|} \cdot \frac{du}{dx}$

3. Thanks. I just didn't know what the derivative of an absolute value function was.

4. Differentiation is not the whole story for this problem, though – there is a critical point at which the function is not differentiable.

A quick sketch of the curve should show that $\displaystyle x^2+4x<0$ for $\displaystyle -3<x<0$ and $\displaystyle x^2+4x>0$ for $\displaystyle 0<x<3$ and $\displaystyle f(x)$ has a nondifferentiable critical point at $\displaystyle x=0.$ Therefore a better solution is to rewrite the function as follows:

$\displaystyle f(x)\ =\ \begin{cases}-x^2-4x & -3<x\leqslant0\\ x^2+4x & 0\leqslant x<3\end{cases}$

You can then use differentiation on the open intervals $\displaystyle (-3,\,0)$ and $\displaystyle (0,\,3)$ separately to find the other critical point(s).