Question

How do you find the extrema for `f(x) = x^2 +2x - 4` for [-1,1]?

Answer 1

Function `f(x)=x^2+2x-4` has in `[-1,1]` a minimum `-5` for `x=-1`.

Explanation:

For any function you can use the first derivative - only a point that suffices `f'(x)=0` can be an extremum. If you're looking for extrema within a given interval you work only with those almost-extrema in the integral.

`f(x)=x^2+2x-4`
`f'(x)=2x+2`
`2x+2=0 => x=-1 in [-1,1]`

In this case the only point which can be considered in finding the extrema is `x=-1` and fortunately it's in the given interval. Now, we evaluate the second derivative and
if `f''(x)>0` then `x` is a local minimum
if `f''(x)<0` then `x` is a local maximum.

`f''(x)=2`
`f''(-1)=2>0` - `x=-1` is a local minimum and f(-1)=-5#.