local maximum & minimum values of a function

[Guest]

I need help solving this function. Also, how do I find the local maximum and minimum values?
f(x)= x^3 - 2x^2 - 8x
I don't even know where to begin.

 

[Unco]

To solve f(x)= x³ - 2x² - 8x

First factor out x:
f(x) = x(x² - 2x - 8)

And factorise the quadratic inside the parentheses.

Relative minima/maxima occur where the derivative:
f'(x) = 3x² - 4x - 8 is zero.

So solve 3x² - 4x - 8 = 0 for x.

To determine their nature (ie. whether they're a relative maximum or minimum), either use the second derivative test (if f''(x)<0, relative maximum, if f''(x)>0, relative minimum), or test the sign of the first derivative, f'(x), before and after each turning point. This should all be covered in your class notes/textbook.

It is an up-down-up cubic so the relative maximum will occur at the lowest turning point, and the relative minimum will occur at the highest.

Edit: Relative is synonymous with local here.