In the previous section, we defined the local maximum and the local minimum and introduced the first derivative test. In this section we are going to work a problem from the first derivative test and introduce the second derivative test.
Suppose we had a function that looked like this: . In order to use the first derivative test we have to be able to find the first derivative which is . Next we need to find the zero of f'. Which is equal to 1/2. To the left of 1/2, f' takes on negative values so we know the function is decreasing on this interval. To the right of 1/2, f' is positive which means the function is increasing. The first derivative test tells us that if f'(c) < 0 to the left of c and f'(c) > 0 to the right of c, then c is a local minumum. We can check our work with a graph of f(x) and f'(x) which are displayed below.
The last topic for this section is the second derivative test which goes something like this. Suppose we have a function f(x) and we know it's second derivative,f''(x). First we set f'(x) equal to zero and solve for x which will give us one or more zeroes of f' that I will call point c's. If f''(c) > 0, then we have a local minimum. If f''(c) < 0, then we have a local maximum. Let's check this rule with our function, . f''(x)=2. So f''(1/2) > 0 for all x. This tells us we have a local minimum at x = 1/2 which is what the first derivative test and the graphs already confirmed.