Steps for Second Derivative Test for Maxima and Minima
Consider a real-valued function f(x) which is defined on a closed or bounded interval [a, b]. Let k be a point in this interval.
In order to conduct the second derivative test on a function f(x), the following steps are followed:
- Differentiate the function f(x) with respect to x to get f'(x).
- Now further differentiate f'(x) with respect to x to get f”(x) i.e. the second derivative of f(x).
- Equate the value of f'(x) to 0 to get the values of x at which f'(x) becomes zero.
- Now calculate the value of f”(x) at points obtained.
On calculating the value of f”(k), we can arrive at the following three conditions:
Case 1: Local Minima
If f'(x) = 0 and k is the required point, then if f”(k) > 0, the point k is said to be the point of local minima.
Case 2: Local Maxima
If f'(x) = 0 and k is the required point, then if f”(k) < 0, the point k is said to be the point of local maxima.
Case 3: Point of Inflection
If f'(k) = 0 and k is the required point, then if f”(x) = 0, the point k is said to be the point of inflection and the function is said to have no point of local maxima and minima.
Learn more about Inflection Point.
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