Calculus Question

Question

Find the absolute maximum and absolute minimum values of f on the given interval.

f(x) = 2x³ − 6x² − 18x + 7,    [−2, 4]

Absolute Minimum= _____

Absolute Maximum= ____

Answer 1

f(x) = 2x^3 -6x^2 - 18x + 7

check the end points of the interval

f(-2) = 2(-8) -6(4) - 18(-2) + 7 =-16 -24 +36 + 7 = - 40 + 43 = 3

f(4) = 2(64) -6(16) - 18(4) + 7 = 128 -96 -72 + 7 = 135-168 = -33

find and check the local extrema and compare them to the f(x) values at the endpoints

take the derivative and set equal to zero

f'(x) = 6x^2 -12x - 18 = 0

x^2 -2x -3 = 0

(x-3)(x+1) = 0

x =-1, 3

plug those values into the cubic function

f(-1) = 2(-1) -6(-1) + 18(-1) + 7 = -2+6-18+7 = -7 = a local minimum > -33

f(3) =2(3) -6(3) + 18(3) + 7 = 6 - 18 +54 + 7 = 67-18 = 49 = a local and absolute maximum

absolute minimum = f(4) = -33 which is the point (4, -33)

absolute maximum = f(3) = 49 which is the point (3, 49)