L'H6PITAL'S RULE
213
25.48
25.49
25.50
25.51
25.52
25.53
25.54
25.55
for any positive integer «.
for any positive integer n.
Let
that of Problem 23.43.
by Problem 24.75. This result generalizes
for any positive integer n.
Hence,
Sketch the graph y = (In x)"/x when n is an even positive integer.
See Fig. 25-1. y' = [n(ln*)'"' - (In*)"]/*
2 = [(Inx)"~\n - \nx)]/x
2 . Setting y' = 0, we find lnx = 0
or n = In x, that is, x = 1 and x = e" are the critical numbers. By the first-derivative test, there is a
relative maximum at x = e", y = (n/e)", and a relative minimum at (1,0). As *-»+», y-*0 by
Problem 25.51. As jc->0
+ , y-»+<».
Fig. 25-1
Fig. 25-2
Sketch the graph of y = (In x)"lx for positive odd integers n.
As in Problem 25.54, x = e" yields a relative maximum. For n > 1, x = 1 is a critical number, but
yields only an inflection point. When n>l, there are two other inflection points. As *-*+<», y—>Q
by Problem 25.51. As x-*0
+ , y-»-<». Figure 25-2 gives the graph for n>l, the graph for n = l is
given in Fig. 24-4.
since
by problem24.74.
Then
Then
by probmel24.75