Question

Use the given function and the given interval to complete parts a and b. f (x )f(x)equals=negative 2 x cubed plus 21 x squared minus 60 x−2x3+21x2−60x on [1 comma 6 ][1,6] a. Determine the absolute extreme values of f on the given interval when they exist. b. Use a graphing utility to confirm your conclusions. a. What​ is/are the absolute​ maximum/maxima of f on the given​ interval? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The absolute​ maximum/maxima is/are nothingat xequals=nothing. ​(Use a comma to separate answers as needed. Type exact​ answers, using radicals as​ needed.) B. There is no absolute maximum of f on the given interval.

Answer 1

(A)The absolute maximum of f(x) is at x=5

Explanation:

F(x)=−2x³+21x²−60x on [1,6]

Find the first derivative.

f'(x)=-6x²+42x-60

Set the first derivative equal to zero.

-6x²+42x-60=0

Solve to find the critical points.

x=2,5

Next, we use the endpoints and all critical points on the interval to test for any absolute extrema over the given interval.

x=1,2,5,6

This is done by evaluating the function at x=1,2,5,6.

F(1)=−2(1)³+21(1)²−60(1)=-41

F(2)=−2(2)³+21(2)²−60(2)=-52

F(5)=−2(5)³+21(5)²−60(5)=-25

F(6)=−2(6)³+21(6)²−60(6)=-36

The maximum will occur at the highest f(x) value and the minimum will occur at the lowest f(x) value.

Absolute Maximum: (5,-25)

Absolute Minimum: (2,-52)

The graph is attached.

The absolute minimum and maximum of f(x) are (2,-52) and (5,-25) respectively.