Find the absolute extrema of the function on the indicated

Yolanda Jorge

Yolanda Jorge

Answered question

2021-12-04

Find the absolute extrema of the function on the indicated interval.
f(x)=xe3x;[1,2]

Answer & Explanation

Twereen

Twereen

Beginner2021-12-05Added 13 answers

Step 1
As a first step, we need to find all the critical points by equating the derivative of f(x) to zero.
We then need to evaluate the integral at all the critical points and the boundary points.
The value of x at which f(x) is maximum will be the global or absolute maxima and the value of x at which f(x) is minimum will be the global or absolute minima
Step 2
f(x)=xe3x
Hence, f(x)=e3x3xe3x=(13x)e3x
For critical points, f'(x) = 0
Hence, (13x)e3x=0
Or, (1-3x)=0 as e3x>0 for any value of
Hence, x = 1/3
Interval boundaries are -1, and 2
Step 3
Hence, f(1)=(1)e3=e3<0
f(13)=13e3×13=e13=0.1226
f(2)=2e3×2=2e6=0.0050
Step 4
Final answers:
Absolute extrema
Absolute minima at x = -1 and the absolute minimum value is e3
Absolute maxima at x = 1/3 and the absolute maximum value is 1/(3e)

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