3. (16 points (8 each). To earn all possible points: Find all of the values whic

Question

3. (16 points (8 each). To earn all possible points: Find all of the values which need to be considered. Evaluate correctly and choose the correct answers.) Use calculus techniques to find the absolute maximum and minimum values of each function over the indicated interval and the x-value(s) at which they occur. Write your answers as "Absolute maximum ofatx or “Absolute minimum ofat x or indicate that none exist. the x-valuets) at which the oc cte that none exst - ?? f(x) = 2x3-3x2-12x-2; (-00,0) f(x)-3-3x 6; [-1,2

Explanation / Answer

a. f(x) = 2x3-3x2-12x-2

f '(x) = 6x2-6x -12

Critical number , f '(x) =0

6(x2-x-2)=0

(x-2)(x+1) =0

x =-1, 2

f(-infinity) = -infinity, f(-1) = -2 -3 +12-2 = 5

f(0) = -2

Absolute minimum of -infinity ia at x=-infinity

Absolute maximum of 5 is at x=-1

b. f(x) = x3-3x +6

f '(x) = 3x2-3

Critical number, f '(x) =0

3(x2-1)=0

x =-1, 1

f(-1) = -1 +3+6 = 8

f(1) = 1-3+6 = 4

f(2) = 8-6+6=8

Absolute maximum of 8 at x=-1, 2 and absolute minimum of 4 at x=1

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