CPM Homework Help : CALC Problem 5-144

Find the coordinates for all of the relative minima, relative maxima and points of inflection for the function $f(x) = (x^2 − 1)(x + 1)^2$ .

Hint 1:

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

Hint 2:

Minima on $f(x)$ happen where $f^\prime(x)$ changes from negative to positive.
Maxima on $f(x)$ happen where $f^\prime(x)$ changes from positive to negative.
Inflection points on $f(x)$ happen where $f^{\prime\prime}(x)$ changes sign.

Hint 3:

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