Find the absolute maximum and absolute minimum values of f

Yolanda Jorge 2021-12-07 Answered
Find the absolute maximum and absolute minimum values of f on the given interval.
f(x)=2x33x212x+1,[2,3]
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Expert Answer

Rosemary McBride
Answered 2021-12-08 Author has 10 answers
Step 1
The function is given by
f(x)=2x33x212x+1,[2,3]
To evaluate : The absolute maximum and minimum of the function on the given interval.
Step 2
Let us evaluate the critical numbers,
dfdx=0
ddx(2x33x212x+1)=0
6x26x12=0
x2x2=0
x22x+x2=0
x(x2)+1(x2)=0
(x+1)(x2)=0
x=1,2 These are critical numbers
Step 3
Now, let us evaluate the values of the function at the critical numbers and at the endpoints of the closed interval [-2,3]
At x=-1,
f(1)=2×(1)33×(1)212×(1)+1=8
Absolute maximum
At x=2,
f(2)=2×(2)33×(2)212×(2)+1=19
Absolute minimum
At x=-2,
f(2)=2×(2)33×(2)212×(2)+1=3
At x=3,
f(3)=2×(3)33×(3)212×(3)+1=8
Step 4
Hence, the absolute maximum and absolute minimum values of the function are :
Absolute maximum value of the function = 8
Absolute minimum value of the function =-19
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