Introduction to Second derivative test

In calculus, two derivative tests are used to discuss a function's behavior. These tests are first-derivative and second-derivative tests. The first derivative test is usually suitable for calculating critical points. In comparison, the second derivative test determines whether the critical issues are local maxima or minima. Let's understand more about the second derivative test and the difference between the first and second derivative tests.

Understanding the Second Derivative Test

The second derivative test is a method to determine the concavity of a function. It calculates the local extreme points of a function under specific conditions. Since this concept is based on a function's rate of change, the second derivative is used. The second derivative of a function is calculated by differentiating the function twice. By definition, the second derivative of a function f(x) is defined as;

$f’’(x)=\frac{d^2}{dx^2}[f(x)]$

Where, 

$\frac{d}{dx}[f(x)]=\lim_{h→0}\frac{f(x+h)-f(x)}{h}$

After applying the first derivative test and calculating the critical point, the 2nd derivative test is used. It determines whether the critical points are local maximum or minimum.

Second derivative test formula

Since the critical points are the points in the domain of a function f(x), where f’(x) is zero or does not exist. These points are analysed by using a second derivative. According to the second derivative test formula, if f(x) is twice differentiable at a critical point x then,

• If f’’(x) < 0, then f has a local maximum at x.
• If f’’(x) > 0, then f has a local minimum at x.
• If f’’(x) = 0, then the second derivative test fails.

If the 2nd derivative test fails, Taylor's theorem can be used sometimes to determine the behaviour of f(x) at that point.

How to do the second derivative test?

A function is differentiated two times to apply a second derivative test for relative extrema. You can also follow the 2nd derivative test steps to simplify calculations. These steps are:

1. Write the function and identify the independent variable. 
2. Calculate the first derivative of f(x), i.e., f’(x) and use the relevant rules according to the type of function. For example, you can use the product rule if there is a product of two functions depending on the same variable.
3. Equate f’(x) to zero to calculate the critical points such as a and b.
4. Calculate the second derivative of f(x) by differentiating its first derivative.
5. Substitute the value of critical points a and b in the second derivative. 
6. If the second derivative of f(x) at a is greater than zero, then a is the local minimum.
7. If the second derivative of f(x) at b is less than zero, then b is the local maxima. 

Let’s understand the implementation of the second derivative test in the following example.

Second derivative test example

Find the maxima and the minima by using the second derivative test of the function, 

$f(x) = x^3 - 12x + 5$

In first step, we will calculate the first derivative, so, 

$f’(x)=\frac{d}{dx}[x^3 - 12x + 5]$

Since the function f(x) contains an algebraic expression with an exponent, therefore, we will use the derivative power rule.

$f’(x)=3x^2-12$

Now to calculate critical points, substitute f’(x) = 0,

$3x^2 – 12 = 0$

$3x^2 = 12$

$x^2 = \frac{12}{3} = 4$

Taking square root on the both sides, we get

$x = ± 2$

Hence we have two critical points 2 and -2. Differentiating f(x) again to get a second derivative.

$f’’(x) = 6x$

Now by using the second derivative, we will calculate f’’(2) and f’’(-2). 

$f’’(2) = 6(2) =12$

$f’’(-2) = 6(-2) = -12$

Since, f’’ (2) > 0, so 2 is the local minima and f’’(-2)<0, so -2 is local maxima.

Comparison between first derivative and second derivative test

The comparison between the higher derivative and second derivative test can be easily analysed using the following difference table.

Conclusion

The second derivative test is a concept of calculus that uses 2nd derivative of a function. It determines the local extreme values of a function that we get from the first derivative of a function. But this test is only applicable when the function is differentiable twice. It fails when the second derivative becomes zero.