What are the absolute extrema of # f(x)= |sin(x) + ln(x)|# on the interval (0 ,9]?

1 Answer
May 21, 2017

Answer:

No maximum. Minimum is #0#.

Explanation:

No maximum
As #xrarr0#, #sinxrarr0# and #lnxrarr-oo#, so

#lim_(xrarr0) abs(sinx+lnx) = oo#

So there is no maximum.

No minimum

Let #g(x) = sinx+lnx# and note that #g# is continuous on #[a,b]# for any positive #a# and #b#.

#g(1) = sin1 > 0# #" "# and #" "# #g(e^-2) = sin(e^-2) -2 < 0#.

#g# is continuous on #[e^-2,1]# which is a subset of #(0,9]#.

By the intermediate value theorem, #g# has a zero in #[e^-2,1]# which is a subset of #(0,9]#.

The same number is a zero for #f(x) = abs(sinx+lnx)# (which must be non-negative fo all #x# in the domain.)