### Home > APCALC > Chapter 5 > Lesson 5.4.1 > Problem5-144

5-144.

Determine the coordinates of all of the relative extrema and points of inflection for the function $f (x) = (x^2 – 1)(x + 1)^2$ .

Max, min, and points of inflection on $f\left(x\right)$ will occur where $f '\left(x\right) = 0$ or DNE or $f ''\left(x\right) = 0$ or DNE. But that alone is not enough to determine which is which.

Minima on $f\left(x\right)$ happen where $f '\left(x\right)$ changes from negative to positive.
Maxima on $f\left(x\right)$ happen where $f '\left(x\right)$ changes from positive to negative.
Inflection points on $f\left(x\right)$ happen where $f ''\left(x\right)$ changes sign.

Another way to distinguish between maxima and minima is to use the 2nd Derivative Test.