What are the local extrema, if any, of #f(x)= 2x^3 – 6x^2 – 48x + 24#?

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Answer 1 · 2016-07-02T12:09:02

local max at x = -2

local min at x = 4

#f(x)= 2x^3 – 6x^2 – 48x + 24#

#f'(x)= 6x^2 – 12x – 48 = 6(x^2 - 2x - 8)#

#= 6 (x-4) (x+2) #

#implies f' = 0# when #x = -2, 4#

#f'' = 12(x - 1)#

#f''(-2) = -36 < 0# ie max
#f''(4) = 36 > 0# ie min

the global max min are driven by the dominant #x^3# term so #lim_{x to pm oo} f(x) = pm oo#

it must look like this..

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