Example 1: Let
(1) First, we rewrite and find So the critical number for f is at x=2. When x>2, so f is increasing in When so f is decreasing in We would expect f to have a minimum at x=2. Since f(2)=1, the minimum point for f is (2,1).
(2) We note that is undefined (and f(2) is defined) so we expect to have a sharp corner or a vertical tangent at x=2. But vertical tangent is impossible, since we already see that f has a min. at x=2. So, it has to be a sharp corner. Pay attention to the following two questions which are related to the slope of the tangent at x=2.
(3) What is (If you know there is a sharp corner and min. at x=2, you could guess the answer to this already). Let's evaluate the limit, (you plug in x=2.0001), you see the denominator is small, and the fraction is posititive, so you get
(4) What is Try this for yourself, you should get
(5) Can you finish the graph? Use plot(((x-2)^(2/3)) + 1,x=-2..4, thickness=3); and use Maple Software to verify your answer, but be cautious, the graph you obtain from Maple is not accurate, because most of computer algebra systems, they have trouble to plot odd root functions. They treat them like even root functions.
(6) Use to find the intervals where f is convave upward and downward, from our graph and analysis, we should see that f is concave downward in Let's see if we get consistent information from Well, since we see that denomator is always positive so the whole fr action is always negative, which implies that f is always concave down.
Example 2: Let . Investigate max/min, concave upward/downward, sharp corner or v ertical tangent and graph the function.
(1) First, and Note that is always positive, so f is always increasing, no max or min.
(2) We note that is undefined but f(-5) is defined (=0), we expect to have a vertical tangent at x=-5 (will not be a sharp corner because we should not have max or min.) Can you guess what and
(3) (Verify).
(4) Since when x<-5, so f is concave upward in When x>-5, so f is concave downward in (-5,
(5) Use plot(4*((x+5)^(3/5)) + 1,x=-2..7, thickness=3); Notice that Maple gives a wrong graph for this problem, do you know why?