Derivative Tests Part II

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.

You can see in the graph of f'(x)=2x-1 the value of f' changes from positive to negative at about 1/2. So that tells us that we have a local minimum, and looking at the graph of f(x) confirms this.

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.