In the last section, we discussed increasing and decreasing functions and looked at their definitions. In this section we are going to discuss derivative tests namely the first derivative test and the second derivative test, but first we need some new definitions.
If for all x within the area c, f(c) f(x), then f has a local minimum at c. If for all x within the area of c, f(c) f(x), then f has a local maximum at c.
The definitions are pretty self-explanatory. If the value of f(c) is less than all points closeby both to the left and to the right, then f has a local minimum at c. Likewise, if the value of f(c) is greater than all the points closeby both to the right and the left, then c is local maximum for f.
If we don't know the graph of f it may help to use the first derivative test. If you've forgotten derivatives, then you can refresh your memory here.The first derivative test says that if f'(c) < 0 to the left of c, and f'(c) > 0 to the right of c, then c is a local minimum. The first derivative test also says something about the local maximum. If f'(c) > 0 to the left of c, and f'(c) < 0 to the right of c, then c is a local maximum.
The first derivative test makes use of the fact that the first derivative tells us the slope of f. In order to have a local minumum, the line must obviously have to slant down initially, and then slant upwards for some amount of distance. If the first derivative is negative, that means we have a downsloping line, and if the first derivative is positive, we have an upward going line. The point in between is the local minumum.
Here we can make use of the intermediate-value theorem which was discussed two sections ago. We will make B a value that is of interest to us. In order for us to find a local minimum we must be able to find where the slope changes from down to up. Since a downslope has a negative first derivative and an upslope has a positive first derivative, there is exactly one point where the first derivative is in between positive and negative and that is where the slope is zero. There exists a zero because the intermediate-value theorem tells us that in the interval [a,c] there is a point b such that f(b) = B, and B is any number between f(a) which is a negative number and f(c) which is a positive number. We want B to equal zero which is between f(a) and f(c). Being able to find where the zeroes of the first derivative are is enormously helpful in finding the local mimimum or local maximum.