let f(x)=2x³ - 12x² - 30x - 6 be defined on -6,6. find: a. the absolute maximum. write your answer as an…

Explanation:

Step1: Find the derivative of (f(x))

Differentiate (f(x)=2x^{3}-12x^{2}-30x - 6) using the power - rule ((x^n)^\prime=nx^{n - 1}). (f^\prime(x)=6x^{2}-24x - 30=6(x^{2}-4x - 5)=6(x - 5)(x+1))

Step2: Find the critical points

Set (f^\prime(x)=0), then (6(x - 5)(x + 1)=0). Solving (x - 5=0) and (x+1=0) gives (x=-1) and (x = 5).

Step3: Evaluate (f(x)) at the critical points and endpoints

Evaluate (f(x)) at (x=-6,x=-1,x = 5,x=6). (f(-6)=2(-6)^{3}-12(-6)^{2}-30(-6)-6=2(-216)-12(36)+180 - 6=-432-432 + 180-6=-708) (f(-1)=2(-1)^{3}-12(-1)^{2}-30(-1)-6=-2-12 + 30-6=10) (f(5)=2(5)^{3}-12(5)^{2}-30(5)-6=2(125)-12(25)-150-6=250-300-150-6=-206) (f(6)=2(6)^{3}-12(6)^{2}-30(6)-6=2(216)-12(36)-180-6=432-432-180-6=-186)

Step4: Determine the absolute maximum and minimum

a. The absolute maximum value of (f(x)) on ([-6,6]) is (10) which occurs at (x=-1), so the ordered - pair is ((-1,10)). b. The absolute minimum value of (f(x)) on ([-6,6]) is (-708) which occurs at (x=-6), so the ordered - pair is ((-6,-708))

Answer:

a. ((-1,10)) b. ((-6,-708))