Second Derivative Test – Explanation, Usefulness, Examples

The Second Derivative is known to measure the rate of change of the first derivative. The sign of the 2nd derivative enables us to know whether the tangent line slope is decreasing or increasing. In this article, I will be discussing the Second Derivative Test and how to solve it.

Keep reading till the end of the article to find out more information about the same!

About the Second Derivative Test

You might ponder on the question “What is the second derivative test? Well, the Second Derivative Test is a form of systematic method of locating the absolute minimum and absolute maximum value of the real-valued function that gets defined on a bounded or closed interval.

The Second Derivative Test can be used for solving optimization problems in engineering, physics, and economics.

Explanation On The Same

The Second Derivative Test is a very systematic method for finding the absolute minimum and maximum of a real-valued function which can either be defined on a closed/bounded interval.

Here, the function f(x) gets defined as a closed interval I, and the “x=k” belongs to the closed interval (I).

The function of f(x), which is actually differentiable at x2 at “x=k” leads to the following three conditions:

x=k is the point of local minima if the value of f'(k) is zero and also the value of f”(k) is >0. The particular point at “x=k” is known as the local minima and the f(k) is referred to as the local minimum value of the f(x).
x=k is a point of local maxima if f'(k) equals the value of ‘0’. It also is a point of local maxima if f” (k)<0. The point at “x=k” is referred to as the local maxima and the f(k) is known as the local maximum value f(x).
If the test happens to fail then f'(k) is going to be equal to zero, and the f”(k) is also going to be equal to the value of zero.

Tip: When practicing you can make use of the second derivative test calculator by Byju’s, to tally if your answer came out to be correct.

Usefulness Of The Test

The Second Derivative Test is beneficial in a whole lot of ways that can also be understood from the following uses:

The Second Derivative Test is also very useful for finding both the minimum and maximum value of a given function. It serves as the optimal solution for any given problem.
This test can be used for finding the local minima and local maxima of a particular function – under specific constraints.
This test also aids in determining whether the concave is down or up.
For the parabolic type of equations, the 2nd Derivative test aids in knowing about the vertex of the parabola or the turning point. It also gives the orientation of the parabola as well.
This derivative test also lets us know about the endpoints of curves.

Example Of The Second Derivative Test

Here is an example of the Second Derivative Test that you can see, to get a better idea about the way of solving it:

Q. Find the minima and maxima of the function(x)=x3-12x+5, by making use of the Second Derivative Test.
AnswerThe given function is f(x)=x3 – 12x + 5.
=> f'(x)=3×2-12=> f'(x)=0; i.e we have x=2 & x-2=> f”(x)=6x=> Therefore, f”(-2)=6x(-2)=-12&f”(-2)<0&x=-2 is Maxima; f”(2)=6×2=12 & f”(2)>0 & x=2 is Minima.

Have A Look : What Is A Voided Check?

Common Queries

Here is a list of some of the more common queries that you might have regarding the topic of “Second Derivative Test”:

Q. Does The Second Derivative Test Always Stand True?

A: The Second Derivative Test is not always true. For instances like the point of inflection, this derivative does not stand true.

If it so happens that the second derivative test is not true then one needs to go back to the first test for verifying the given values.

Q. What Is The Difference Between First And Second Derivative Tests?

A: The first and second derivative test has a major difference, while both of them are truly helpful in searching the local maximum and minimum points.

The first derivative is known to only take the first derivative of a given function. While the second derivative takes both the first and second derivative of any given function.

Q. Why Does The Second Derivative Test Seem To Fail?

A: The Second Derivative Test seemingly fails for two instances. If for a given function which is f(x), the 2nd does not exist f” (x)=0.

Also for the value of “x=k”, if the value of the 2nd Derivative stands at f'(k)=0, then the function fails.

To Wrap It Up!

There’s no need to fret about finding derivatives. Even though finding them might sound a bit tough at first, they are really easy to find once you get the hang of it.

Thank you for reading till the end of this particular article. I hope you found the information regarding the “Second Derivative test” useful.

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