Second Derivative Test

What is Second Derivative Test

In mathematics, the meaning of the second derivative stands for a function which is the derivative of the derivative of that function. Would you want to know how to write a second derivative in mathematical expression? Write it as: - f 00(x) or as d 2 f dx2. Now do you know the utility of the first derivative with respect to the second derivative? While the first derivative can make us aware if the function is increasing or decreasing, the second derivative puts into the picture if the first derivative is increasing or decreasing.

Conditions of Concavity for Second Derivative Test

Always keep in mind that, if the 2nd derivative is positive, it states that the first derivative is increasing, so that the slope of the line of tangent to the function is increasing as x increases. We experience this occurrence graphically as the curve of the graph being concave up, that is, fashioned like a parabola opening upward.

Now, in the similar vein, if the second derivative comes about as negative, then the first derivative is decreasing, in order as the slope of the tangent to the function is decreasing as ‘x’ increases. Illustratively in Graphs, we notice this as the curve of the graph which is concave down, that is, modeled like a parabola opening downward. At the points where the second derivative is 0, we do not acquire knowledge of anything with respect to the shape of the graph: it may either be concave up or concave down, or it may be changing all- through concave up to concave down or vice-versa. Hence, to sum up:

If d 2 f dx2 (p) is greater than 0 at x = p, then f(x) is concave up at x = p.

If d 2 f dx2 (p) is lesser than 0 at x = p, then f(x) is concave down at x = p.

If d 2 f dx2 (p) is 0 at x = p, then we are unaware of anything new about the attitude of f(x) at x = p. 

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Utility of Second Derivative Test

The second derivative test is factually less dominant than the first derivative test. That clearly made you curious as to why then is it ever used? A principal reason is that in conditions where it is conclusive, the second derivative test is commonly and comparatively easier to apply. This, in turn, is due to the reason that the second derivative test solely needs the calculation of formal expressions for derivatives. As well, it requires the assessment of the symbols of these expressions at preferably a point than on an interval. Assessments at a point usually necessitate less arithmetic/ algebraic maneuver or handling.

Moreover, a 2nd derivative test can help identify whether a stationary point is a Local Maxima or a Local Minima or if it is a global maxima/global minima. It is found out by comparing the value of local maxima/minima with other global maxima/global minima.

Usability of Second Derivative Test

The second derivative test is often most useful when seeking to compute a relative maximum or minimum if a function has a first derivative that is (0) at a particular point. Since the first derivative test is found lacking or fall flat at this point, the point is an inflection point. The second derivative test commits on the symbol of the second derivative at that point. If it is negative, the point is a relative maximum, whereas if it is positive, the point is a relative minimum.

Solved Examples

Find and use the second derivative of a function

Take f(x) = 3x 3 − 6x 2 + 2x − 1.

Now,

 f 0 (x) = 9x 2 − 12x + 2, and f 00(x) = 18x − 12.

That being so, at x = 0,

The 2nd derivative of f(x) is −12,

So we have an understanding that the graph of f(x) is concave down at x = 0.

Similarly, at x = 1, the 2nd derivative of f(x) is f 00(1) = 18 1 − 12 = 18 − 12 = 6,

Thus, the graph of f(x) rests at concave up at x = 1

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Did You Know 

  1. There is also a one-sided version of 2nd derivative test

  2. a one-sided version works as an alternative or say a remedial option for cases to not revert to the first derivative test. 

  3. If the one-sided derivatives of f' is available at c, then we can check that both one-sided derivatives of f' have the sign for f'' set forth 

FAQ (Frequently Asked Questions)

Q1. Can Maxima and Minima be Used in day-to-day Applications?

The maximum and the minimum values can be used in different fields such as engineering and researching. Apart from the benefits of learning the working rule of maxima and minima in your course curriculum, if you aspire to become an engineer or doctor, it will have sound advantages.    

So, let’s take a look of how and where maxima and the minima can be used. Check as follows:-

To an Economist – The maximum and the minimum values of the total revenue generated function can be utilized to attain an idea of the limits the company must put on production, procurement or operations as to not run out of cash flow.

To an Engineer – The maximum and the minimum values of a function can be utilized to identify its frontier in real-life. For example, if you can determine an appropriate function for the distance traveled by a loading truck; then finding the maximum possible distance traveled by truck can help you choose the tier-of-tyres required or deciding the weight carrying capacity that would withstand the speeds, for the truck to run smoothly.

To a Doctor – The maximum and the minimum values of the function narrating the total sugar level in the bloodstream can be used to choose the dosage the doctor needs in order to prescribe to different patients so as to bring their sugar levels to normal.

Q2. How to Find the Second Derivative?

The second derivative of an unstated function can be discovered taking into account the sequential differentiation of the initial equation F(x,y)=0. As the first step, we obtain the first derivative in the expression y′=f1(x,y). In the next step, we determine the second derivative, which can be demonstrated in terms of the variables x and y as y′′=f2(x,y).

Q3. What are the Various Types of Maxima and Minima?

The maxima or minima can also be termed as extremum which refers to the extreme value of the function. For example, take a function y = f(x) stated on a known domain of x. Depending upon the interval of x, on which the function achieves an extremum, the extremum can be called as a ‘local’ or a ‘global’ extremum.