Derivatives can be used as a tool for maximizing or minimizing the output of a function.
The first derivative test is a method for finding local extrema of a function at a critical point.
The first derivative test is performed by checking the first derivative right before the point being checked and right after. If the derivative switches from positive to negative there is a local maximum there, alternatively if the derivative switches from negative to positive there is a local minimum there. If the sign of the derivative stays the same there is no extrema at that point.
Given the function
is shown in blue, with critical points shown as red dots. Additionally the derivative of is also shown in green.
| 0 | |||||
| Decreasing | Minimum | Increasing | Maximum | Decreasing |
Tables like the example above are a good way of finding all the local extrema on an interval.
There are 2 local extrema, 1 local minimum at
The second derivative test is a method for potentially finding local extrema of a function at a critical point. The second derivative test can sometimes be inconclusive in which case the first derivative test must be used.
The second derivative test is performed by checking the second derivative at a critical point.
The test can result in 3 different cases
Given the function
is shown in blue, with critical points shown as red dots.
There are 2 local extrema, 1 local minimum at
Because neither of these second derivatives were 0 we know that the second derivative test was conclusive so we don't need to use the first derivative test.