How do use the first derivative test to determine the local extrema $f (x) = - x^{3} + 12 x$ $f(x)= -x^3 + 12x$ ?

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Answer 1 · 2015-09-13T13:10:07

Refer to explanation

To find the local extrema we need the zeroes of the derivative of f

Hence we have that

$f'(x)=0=>-3x^2+12=0=>-x^2+4=0=>x=+-2$

Now we see how f'(x) behaves around the zeroes

graph{-x^2+4 [-10, 10, -5, 5]}

We see at x=-2 f'(x)>0 and at x=2 f'(x)<0

Hence at x=2 local maxima f(2)=16 and at x=-2 local minima f(-2)=-16

graph{-x^3+12x [-40, 40, -20, 20]}

How do use the first derivative test to determine the local extrema f(x)= -x^3 + 12xf(x)=−x3+12xf(x)= -x^3 + 12x?