# Differential Equations/local extrema

Question
hello,

I have a question.

y=g(X)
domain all real number and differentiable every where

g'(-3)=g'(4)=g'(8)=0
g"(-3)=0
g"(4)=3
g"(8)=-6

whats the local extrema

I am confused

i think
Local Max is -6 at x=8
Local Min is 3 at x=4

but what does g'(-3)=g'(4)=g'(8)=0 means then.

Thanks
Mnsor

If f'(x) = 0, the point is a maximum, minimum, or inflection points.
If f"(x) > 0, f(x) is a local minimum.  If f"(x) < 0, f(x) is a local maximum.
If f"(x) = 0, there is an inflection point in f() at x.

Since g'(-3) = 0, g'(4) = 0, and g'8) = 0, x=-3, x=4, and x=8 are all critical points.
Since g"(-3) = 0, that means there is an inflection point in g(x) at x = -3.
Since g"(4) = 3, and 3>0, that means there is a minimum in g(x) at x = 4.
Since g"(8) = -6, and -6<0, that means there is a maximum in g(x) at x = 8.

The values of g(x) at any of these points are not given.

Differential Equations

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#### Scott A Wilson

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I am capable of solving most ordinary differential equations and a few not so ordinary, but those are few and far between.

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