What are the absolute extrema of # f(x)= 6x^3 − 9x^2 − 36x + 3 in [-4,8]#?

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Answer 1 · 2018-06-09T18:35:23

# (-4,-381) # and # (8,2211) #

In order to find the extrema, you need to take the derivative of the function and find the roots of the derivative.

i.e. solve for # d/dx [f(x)] = 0 # , use power rule:

#d/dx [6x^3 - 9x^2-36x+3 ] = 18x^2-18x-36 #

solve for the roots:
# 18x^2-18x-36 = 0 #
# x^2-x-2 = 0 # , factor the quadratic:
# (x-1)(x+2) = 0 #
# x = 1, x = -2 #

# f(-1) = -6-9+36+3 = 24 #
#f(2) = 48-36-72+3 = -57 #

Check the bounds:
# f(-4) = -381 #
# f(8) = 2211 #

Thus the absolute extrema are # (-4,-381) # and # (8,2211) #