extreme values: f (x) = 1/x + ln x 0.5(</=) x (</=) 4

[tsh44]

Hello can someone please outline how to find the extreme values of a function analytically using the following problem. I know the answers but are not sure how to arrive at them.

f (x) = 1/x + ln x 0.5(</=) x (</=) 4

The book says to find the derivative which is -1/x^2 + 1/x. But what does this tell you? The max value answer is 1/4 + ln 4 at x= 4. the min value is 1 at x=1.
the local max is (.5, 2- ln2). thanks so much. i have a test tomorrow and I am lost

 

[arthur ohlsten]

f)x)=1/x + ln x for .5<=x<=4
take derivative to determine max or min

f'(x)=-x^-2 +1/x set it =0
0=1/x[1/x+1] either
1/x=0 x=+/-00 not in domain of .5 to 4

[1/x +1]=0
x=-1 this is not in f(x) domain of .5 to 4

The maximum value is found at the min or max value of the domain .5<x<.4
x=1/2 F(1/2)=2-ln2

x=4 f(4)= 1/4 +ln4 maximum value in domain of .5 to 4

Arthur