Consider the function f(x)=x-3\ln (x) on the interval [\frac{1}{3}, 7]. The

smismSitlougsyy

smismSitlougsyy

Answered question

2021-12-03

Consider the function f(x)=x3ln(x) on the interval [13,7].
The absolute maximum value is
and this occurs at x equals
The absolute minimum value is
and this occurs at x equals

Answer & Explanation

Louis Gregory

Louis Gregory

Beginner2021-12-04Added 14 answers

Step 1
Consider the given function,
f(x)=x3ln(x)
differentiate the function with respect to x,
f(x)=ddx(x3ln(x))
=ddx(x)3ddx(ln(x))
=13(1x)
=13x
find the critical points substituting f′(x)=0,
13x=0
3x=1
x=3
Step 2
use the second derivative test at x=3
f(x)=ddx(13x)
=3x2
f(3)=3(3)2
=13
since at x=3, f''(x)<0 so the function has maxima at x=3.
so the absolute maximum value of the function is,
f(3)=33ln(3)
=3-3.295
=-0.295

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