108 STEP 4. Review the Knowledge You Need to Score High
7.2 Determining the Behavior of Functions
Main Concepts:Test for Increasing and Decreasing Functions, First Derivative Test and
Second Derivative Test for Relative Extrema, Test for Concavity and
Points of InflectionTest for Increasing and Decreasing Functions
Letf be a continuous function on the closed interval [a,b] and differentiable on the open
interval (a,b).- If f′(x)>0on(a,b), thenf is increasing on [a,b].
- If f′(x)<0on(a,b), thenf is decreasing on [a,b].
- If f′(x)=0on(a,b), thenf is constant on [a,b].
Definition:Let f be a function defined at a numberc. Thencis a critical number off if
eitherf′(c)=0orf′(c) does not exist. (See Figure 7.2-1.)Figure 7.2-1
Example 1
Find the critical numbers off(x)= 4 x^3 + 2 x^2.
To find the critical numbers of f(x), you have to determine where f′(x)=0 and where
f′(x) does not exist. Notef′(x)= 12 x^2 + 4 x, and f′(x) is defined for all real numbers. Let
f′(x)=0 and thus 12x^2 + 4 x=0, which implies 4x(3x+1)= 0 ⇒x=− 1 /3orx=0.
Therefore, the critical numbers off are 0 and− 1 /3. (See Figure 7.2-2.)[–1,1] by [–1,1]
Figure 7.2-2