Find where the function

f(x) = 3x⁴12x³108x²+8

is increasing and where it is decreasing. SOLUTION

f'(x) = 12x³36x²216x= 12x

x

x+

To use the I/D Test, we have to know where

f'(x) > 0

and where

f'(x) < 0.

This depends on the signs of the three factors of

f'(x),

namely, 12x,

x ,

and

x+ .

We divide the real line into intervals whose endpoints are the critical numbers (smallest), 0 and (largest) and arrange our work in a chart. A plus sign indicates that the given expression is positive, and a negative sign indicates that it is negative. The last column of the chart gives the conclusion based on the I/D Test. For instance,

f'(x) < 0

for

0 <x<6,

sofis ---Select--- decreasing increasing on

(0,6).

(It would also be true to say thatfis decreasing on the closed interval

[0,6].)

Interval	12x	x6	x+3	f'(x)	f
x<3					decreasing on (,3)
3<x< 0			+	+	---Select--- decreasing increasing on (3, 0)
0 <x<6	+		+		decreasing on (0,6)
x>6	+	+	+	+	---Select--- decreasing increasing on (6,)

The graph offshown in the figure confirms the information in the chart.