I have a question.
domain all real number and differentiable every where
whats the local extrema
I am confused
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.
A function g(x) has critical points at g'(x) = 0
We can use the second differential to classify the type of extrema we have at any such point.
Now, according to the information provided we can see that there are critical points at x = -3, 4 and 8. The second differential thus confirms the nature of the extrema at the last two points like you have stated since we know that;
A local maximum occurs when g'(x) = 0 and g''(x) < 0
A local minimum occurs when g'(x) = 0 and g''(x) > 0
But when g'(x) = 0 and g''(x) = 0, then no clear conclusion can be made as in the cases of x^3, x^4 and -x^4 where we would have a point of inflection, minimum and maximum points respectively.
Back to the statement g'(-3) = g'(4) = g'(8) = 0, it is simply declaring that g'(x) = 0 at each of the related points which is the criteria for them being critical points in the first place (otherwise the second differential test would be meaningless).