Test for increasing or decreasing functions
Let f be differentiable on the interval (a,b).
Definition of a critical number
If f is defined at c, then c is a critical number of f if or is undefined at c.
Use of the critical numbers
Test for finding the relative extrema
Suppose that f has a critical number at x=c and .
Examples
Find the critical numbers and open intervals on which the function is increasing or decreasing.
Example: If f(x)=-2x2+4x+3 (page 180, #19). First we find By setting we find x=1, which is the critical point of f.
(1) If x>1 (say x=2), then thus f is decreasing in
(2) If x<1 (say x=0), then thus f is increasing in
(3) f has a relative maximum at x=1. (note the maximum point on the graph will be (1,f(1))=(1,5).)
Example: If f(x)=3x3+12x2+15x (page 180, #21). First we find By setting we find and x=-1.
(1) The critical numbers are and x=-1.
(2) The signs of is displayed as follows: Therefore, f is increasing on and decreasing on
(3) There is a relative maximum at and a relative minimum at x=-1. Use MFI to check your answer.
Example: If First
Notice that h(1) is defined but is undefined, so x=1 is a critical number of h.
If x>1, then If x<1, then Thus h is increasing everywhere. There is no relative extrema.
Example: If First, Notice that the crical numbers are at x=2 and x=3.
(1) If say x=1,
(2) If say x=2.5,
(3) If say x=4, is undefined, also notice that the domain of f does not include So we don't have graph for x>3.
Therefore, f is increasing on and decreasing (2,3).
There is a relative maximum at x=2. Use plot(2*x*sqrt(3-x),x=-5..5, thickness=3); to obtain a plot from MFI.
Example: If We see
Notice that the critical numbers of f are at x=-1,1; x=0 is not a critical number because f(0) is undefined even if is undefined.
However, when we determine the sign of we need to take x=0 into consideration.
Therefore, f is increasing on and decreasing on (-1,1).
f has a relative maximum at x=-1 (be sure that f(-1) is defined), and a relative minimum at x=1.