Second Derivative Test
The second derivative test is a systematic method of finding the absolute maximum and absolute minimum value of a realvalued function defined on a closed or bounded interval. The second derivative test can be used in solving optimization problems in physics, economics, engineering.
Let us learn more about the second derivative test, steps for the test, uses, and examples on 2nd derivative test.
What Is Second Derivative Test?
The second derivative test is a systematic method of finding the absolute maximum and absolute minimum value of a realvalued function defined on a closed or bounded interval. Here we consider a function f(x) defined on a closed interval I, and a point x= k belongs to a closed interval (I). Here we consider a function f(x), which is differentiable twice at x = k, then we have the following three conditions.
 x = k, is a point of local maxima if f'(k) = 0, and f''(k) < 0. The point at x= k is the locl maxima and f(k) is called the local maximum value of f(x).
 x = k is a point of local minima if f'(k) = 0, and f''(k) >0 . The point at x = k is the local minima and f(k) is called the local minimum value of f(x).
 The test fails if f'(k) = 0, and f''(k) = 0. And the point x = k is called the point of inflection.
Here if the test fails at point x=k, we go back to the first derivative test and once again check if it is the local maxima or the local minima.
Steps for Second Derivative Test
The following sequence of steps facilitates the second derivative test, to find the local maxima and local minima of the realvalued function.
 Find the first derivative f'(x) of the function f(x)
 Equalize the first derivative to zero f'(x) = 0 and find the limiting points \(x_1, x_2\).
 Find the second derivative of the function f''(x).
 Substitute the limiting points in the second derivative\(f''(x_1), f''(x_2)\)..
 If the second derivative is greater than zero\(f''(x_1) > 0\), then the limiting point \((x_1)\) is the local minima.
 If the second derivative is lesser than zero \(f''(x_2)<0\), then the limiting point \((x_2)\) is the local maxima.
Uses of Second Derivative Test
The second derivative test is beneficial in a number of ways, which can be understood from the following uses.
 The second derivative test can be used to find the local maxima and local minima of a function, under certain constraints.
 The second derivative test is useful to find the maximum or minimum value of the function, which gives the optimal solution for the problem situation.
 For a parabolic equation the second derivative test helps in knowing the turning point or the vertex of a parabola, and also gives the orientation of the parabola.
 The second derivative test helps in knowing the extreme points of the curves.
 The second derivative test helps us to know if the curve is concave up or concave down.
Further, the second derivative test can be supposed to be useful in the following example situations.
 The profit from a grove of orange trees is given by the expression P(x) = ax + bx^{2}+ cx^{3} + d, where a, b are constants and x is the number of mango trees per acre. To find the number of mango trees per acre required to maximize the profit we use this second derivative test.
 A ball thrown in the air from the top of a building of height 10m, travels along the path given by the formula h(x) = 60 + x  x^{2}/60., where x is the horizontal distance and h(x) is the height of the ball. To find the maximum height the ball would reach, we use the second derivative test.
 A helicopter of the enemy is traveling along the path defined by the equation P(x) = x + 7, and a soldier placed at the point (1, 2) wants to hit the helicopter. Here to find the minimum distance at which the soldier can hit the helicopter, we can use the second derivative test.
Related Topics
The following topics help for a better understanding of the second derivative test.
Examples on Second Derivative Test

Example 1: Find the maxima and the minima of the function (x) = x^{3}  12x + 5, by using the second derivative test.
Solution:
The given function is f(x) = x^{3}  12x + 5
f'(x) = 3x^{2}  12
f'(x) = 0, Hence we have x = 2, and x = 2
f''(x) = 6x
f''(2) = 6 x 2 = 12 , and f''(2) > 0 , and x = 2 is the minima
f''(2) = 6 x (2) = 12, and f''(2) < 0, and x = 2 is the maxima.
Therefore by using the second derivative test, the local maxima is 2, with a maximum value of f(2) = 21, and the local minima is 2, with a minimum value of f(2) = 11.

Example 2: Find the local maxima and local minima of the function f(x) = x^{3}  6x^{2}+9x + 15. using the second derivative test.
Solution:
The given function is f(x) = x^{3}  6x^{2}+9x + 15.
f'(x) = 3x^{2}  12x + 9
f'(0) = 3(x^{2}  4x + 3)
x^{2}  4x + 3 = 0 or (x  1)(x  3)=0.
Here x = 1, and x = 3
f''(x) = 6x  12
f''(1) = 6(1)  12 = 6  12 = 6., f''(1) < 0, and x = 1 is the maxima.
f''(3) = 6(3)  12 = 18  12 = 6, f''(3) > 0, and x = 3 is the minima.
Therefore by using the second derivative test, the local maxima is 1, with a maximum value of f(1) = 19, and the lock minima is 3, with a minimum value of f(3) = 15
FAQs on Second Derivative Test
What Is Second Derivative Test?
The second derivative test is a systematic method of finding the local maximum and minimum value of a function defined on a closed interval. Here we consider a function f(x) defined on a closed interval I, and a point x= k in this closed interval. The following are the three outcomes of the second derivative test.
 x = k, is a point of local maxima if f'(k) = 0, and f''(k) < 0. The point at x= k is the locl maxima and f(k) is called the local maximum value of f(x).
 x = k is a point of local minima if f'(k) = 0, and f''(k) >0 . The point at x = k is the local minima and f(k) is called the local minimum value of f(x).
 The test fails if f'(k) = 0, and f''(k) = 0. And the point x = k is called the point of inflection.
How Do You Do the Second Derivative Test?
The following sequence of steps facilitates the second derivative test, to find the local maxima and local minima of the realvalued function.
 Find the first derivative f'(x) of the function f(x), and equalize it to zero f'(x) = 0 and find the limiting points \(x_1, x_2\).
 Find the second derivative of the function f''(x), and substitute the limiting points.
 If the second derivative is greater than zero\(f''(x_1) > 0\), then the limiting point \((x_1)\) is the local minima, and If the second derivative is lesser than zero \(f''(x_2)<0\), then the limiting point \((x_2)\) is the local maxima.
What Is the Difference Between First Derivative Test and Second Derivative Test?
Is the Second Derivative Test Always True?
The second derivative test is not always true. For situations of point of inflection, it does not hold true. If the second derivative test is not true, we go back to the first derivative test and verify for the given values.
Why Does the Second Derivative Test Fail?
The second derivative test fails in two instances. If for a given function f(x), the second derivative does not exist f''(x) = 0, then the test fails. Also for a value x = k, if the second derivative value f'(k) = 0, then too the function fails.
How Do You Find the Maximum and Minimum of Second Derivative Test?
The second derivative test is a systematic method of finding the maximum and minimum value of a closed value function defined on a closed or bounded interval. Here we consider a function f(x) defined on a closed interval I, and a point x= k belongs to a closed interval (I). The maximum and minimum value can be understood through the below points.
 Maximum Value: x = k, is a point of local maxima if f'(k) = 0, and f''(k) < 0. The point at x= k is the locl maxima and f(k) is called the local maximum value of f(x).
 Minimum Value: x = k is a point of local minima if f'(k) = 0, and f''(k) >0 . The point at x = k is the local minima and f(k) is called the local minimum value of f(x).
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