CHAPTER 25
214
25.56
Graph y = x"e * for positive even integers n.
See Fig. 25-3. v' = -x"e~' + nx^e" = x
n '
l
e -'(n - x). Hence, x = n and x = 0 are critical numbers. The first-derivative test tells us that there is a relative maximum at x = n and a relative minimum
by Problem 24.75. As
As
at x = 0.
Fig. 25-3
25.57 Graph y = x"e " for odd positive integers n.
As in Problem 25.56, there is a relative maximum at x = n. For n > 1 [Fig. 25-4(a)], calculation of the
second derivative yields x"~
2 e~*[x
2 — 2nx + n(n — I)]. Then, there is an inflection point at x = 0, and two
other inflection points in the first quadrant. For the special case n = 1 [Fig. 25-4(fc)], y" = e~'(2 — x), and
there is only one inflection point, at x = 2. In either case, as *-»+<», y-»0, and, as jc-»-x,
y—» —oo.
Fig. 25-4