Question

Why use the second derivative to find min/max points?

Answer 1

Couldn't we just input a value in the first derivative that is close to a critical point from both the positive and negative sides. If the first derivatives gives a negative value for the side of x that is more negative, and the first derivative gives a value that is positive for the side of x that is more positive, then don't we know that critical point is a minimum value? b/c the original function's slope is negative approaching from negative infinity and positive approaching from positive infinity?

Answer 2

Yes you can do exactly what you just explained, but the second derivative only needs to be proved using one point for reference (the critical point itself). If it is negative at the critical point in the second derivative, then the function is curving downwards at that point, like a mountain (maximum), and vice versa for the negative case.

Answer 3

Thanks