Find the absolute maximum and minimum values of f on the given interval. f(x)=4x^{3}-6x^{2}-24x+9. [-2,3]

Lennie Carroll

Lennie Carroll

Answered question

2021-05-09

Find the absolute maximum and minimum values of f on the given interval.
f(x)=4x36x224x+9.[2,3]

Answer & Explanation

ottcomn

ottcomn

Skilled2021-05-10Added 97 answers

Step 1
We will find the first derivative using the power rule
f(x)=4x36x224x+9
f(x)=4(3x2)6(2x)24
f(x)=12x212x24
Step 2
We find the critical values by solving f(x)=0
12x212x24=0
12(x2x2)=0
12(x2)(x+1)=0
x=2, 1
Then we find the values of the function at the critical points and at the endpoints.
f(x)=4x36x224x+9
f(2)=4(2)36(2)224(2)+9=1
f(1)=4(1)36(1)224(1)+9=23 (max)
f(2)=4(2)36(2)224(2)+9=31 (min)
f(3)=4(3)36(3)224(3)+9=9
Answer:
Absolute minimum value= -31
Absolute maximum value=23

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