How do you find the local max and min for f(x)=2x^3 + 5x^2 - 4x - 3f(x)=2x3+5x24x3?

1 Answer
May 11, 2018

x=-2x=2 is a local maximum, and x=1/3x=13 is a local minimum.
See explanations below.

Explanation:

f(x)=2x³+5x²-4x-3
What we know is that there's a local extremum when f'(x)=0
f'(x)=6x²+10x-4
6x²+10x-4=0
6(x²+5/3x-2/3)=0
x²+5/3x-2/3=0
x²-x/3+2x-2/3=0
x(x-1/3)+2(x-1/3)=0
(x+2)(x-1/3)=0
Now we can clearly see that when x=-2 and x=1/3, f'(x)=0
Also, we know that out of roots, f(x)=ax³+bx²+cx+d take the sign of -a in -oo and the sign of a in +oo. Because of that, we can deduce that x=-2 is a local maximum, and x=1/3 is a local minimum.
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