EXAMPLE 2 Find the local minimum and maximum values of the function below.

f(x) = 3x⁴ 12x³ 168x² + 4

Video Example SOLUTION

f'(x) = 12x³ 36x² 336x = 12x(x 7)(x + 4)

From the chart

Interval	12x	x 7	x + 4	f'(x)	f
x < 4					decreasing on (, 4)
4 < x < 0			+	+	increasing on (4, 0)
0 < x < 7	+		+		decreasing on (0, 7)
x > 7	+	+	+	+	increasing on (7, )

we see that

f'(x)

changes from negative to positive at

4,

so

f(4) =

is a local minimum value by the First Derivative Test. Similarly,

f'(x)

changes from negative to positive at 7, so

f(7) =

is a local minimum value. And,

f(0) =

is a local maximum value because

f'(x)

changes from positive to negative at 0.