The second derivative test is useful when trying to find a relative maximum or minimum if a function has a first derivative that is zero at a certain point. Since the first derivative test fails at this point, the point is an inflection point. The second derivative test relies on the sign of the second derivative at that point. If it is positive, the point is a relative minimum, and if it is negative, the point is a relative maximum.
I want to talk about another method for finding relative max and min called the second derivative test and here is the test right here. The second derivative test for relative maxima and minima, let f be a function and c a point in its domain. If the first derivative of f at c is 0 and the second derivative is positive then f has a relative minimum at x=c and if f prime of c is 0 and f double prime is a negative then f has a relative maximum at x=c and that seems kind of random so let's look at a picture and see why that's true. This is the first case f prime of c is 0, remember that that indicates that you've got a horizontal tangent there, so the graph is going to level of. If you also have that f double prime is positive it means that the curve is concave up at that point and that is what indicates a relative min.
The other case where f prime of c is 0 and f double prime is negative, here you've got the graph is concave down and so that's going to indicate a relative max. So the second derivative test does the same thing that the first derivative does. It shows you whether or not you have relative max or min so why do you need 2 tests? Well first of all the second derivative test is sometimes easier to implement than the first derivative test. So why do we always use the second derivative test? The answer to that is that it doesn't always work so when the second derivative test doesn't work you need to use the first derivative test. But the second derivative test is sometimes easier.