First and Second Derivative Test
The first derivative test and second derivative test both are used to find local maxima and minima but there is certain difference between the two. The following table lists the key differences between both the tests.
| Test | First Derivative Test | Second Derivative Test |
|---|---|---|
| Critical Point | Where f'(x) = 0 or f'(x) is undefined | Where f'(x) = 0 or f'(x) is undefined |
| Local Minima | f'(x) changes from negative to positive | f”(x) > 0 at the critical point |
| Local Maxima | f'(x) changes from positive to negative | f”(x) < 0 at the critical point |
| Neither Minima nor Maxima | f'(x) fails to exit. | f”(x) = 0 (inconclusive) |
Second Derivative Test
Second Derivative Test is one of the methods in calculus to find the maxima and minima of a curve. Other than, the second derivative test there is also a first derivative, which can be referred to as a rudimentary version of the second derivative test.
First derivative test helps us find critical points for a given function but does not tell us about the nature of the function at these points. We also come across cases where we cannot get critical points as the first derivative test fails. Second derivative test is used in these cases. The second derivative test clearly tells us if the critical point obtained is a point of local maximum or local minimum. Second derivative test is also helpful in solving various problems in different fields such as science, physics, and engineering. In this article, we shall discuss the second derivative test in detail.
Table of Content
- What is Second Derivative Test?
- Steps for Second Derivative Test for Maxima and Minima
- Examples of Second Derivative Test
- Uses of Second Derivative Test
- Difference between First and Second Derivative Test
- Multivariable Second Derivative Test