Determine the location and value of the absolute extreme value of f on the given interval, if they exist. f(x)=x\ \ln \frac{x}{5}\ on\ \left[0.1, 5\right]

Cem Hayes 2020-11-17 Answered
Determine the location and value of the absolute extreme value of f on the given interval, if they exist.
f(x)=x lnx5 on [0.1,5]
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Expert Answer

tafzijdeq
Answered 2020-11-18 Author has 92 answers

Step 1
Given Data
The function is f(x)=xln(x)5.
The interval is [0.1,5].
Differentiate the function f(x)=xln(x)5 with respect to x and equate to zero to evaluate the location of the absolute extreme,
ddx(f(x))=ddx(xln(x)5)
fprime(x)=ln(x)5ddx(x)+xddx(ln(x)5)
0=ln(x)51+x(1x)(15)
ln(x)5+15=0
ln(x)+15=0
ln(x)=1
x=e1
x=1e
The value of absolute extreme x=1e lie in given interval [0.1,5].
Hence the location of the absolute extreme value of f on the given interval is x=1e.
Step 2
Evaluate the function f(x)=xln(x)5 by substituting the value x=1e,
f(x)=xln(x)5
f(1e)=1eln(1e)5
=0.0735
<0 Hence the minimum value of the absolute extreme value of f on the given interval is -0.0735, which is less than zero.

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Jeffrey Jordon
Answered 2021-11-03 Author has 2047 answers

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