Maximum Value Function

 Maximum value of a function is either a relative (local) maximum or an absolute (global) maximum value of a function. Relative (local) maximum is the highest point in a particular section of a graph. Absolute (global) maximum is the highest point over the entire domain of a function or relation. The first derivative and the second derivative are common methods used to find maximum values of a function. 

 

Maximum Value of a Function – Examples

 

Example 1: Find the maximum value of a function; y = 14x2 + 14x + 5.

Solution:

y = 14x2 + 14x + 5

Derivate the equation

dy/dx = 28x + 14 = 0

2x + 1 = 0

x = -1/2

Put the value of x into above equation

y = 14(-1/2)2 + 14(-1/2) + 5

y = 3.5 - 7 + 5

y = 1.5

Therefore maximum value of a function is 1.5

Example 2: Find the maximum value of a function; y = 2x3 + 4x2 – 5x + 5.

Solution:

y = 2x3 + 4x2 – 5x + 5

Derivate the equation

y' = 6x2 + 8x - 5 = 0

y’’ = 12x + 8 = 0

3x + 2 = 0

x = -2/3

Put the value of x into above equation

y = 2(-2/3)3 + 4(-2/3)2 – (5(-2/3)) + 5

y = -0.59 + 1.7 + 3.3 + 5

y = 9.41

Therefore maximum value of a function is 9.41

 

Example 3: Find the maximum value of a function; y = 12x2 + 3x + 6.

Solution:

y = 12x2 + 3x + 6

Derivate the equation

dy/dx = 24x + 3 = 0

8x + 1 = 0

x = -1/8

Put the value of x into above equation

y = 12(-1/8)2 + 3(-1/8) + 6

y = 0.1875 – 0.375 + 6

y = 5.81

Therefore maximum value of a function is 5.81

 

Example 4: Find the maximum value of a function; y = 7x4 - 2x3 – 4x2 – 5x + 4.

Solution:

y = 7x4 - 2x3 – 4x2 – 5x + 4

Derivate the equation

y’ = 28x3 – 6x2 – 8x - 5 = 0

y’’ = 84x2 - 12x - 8 = 0

y’’’ = 168x – 12

y = 14x -1 = 0

x = 1/14

Put the value of x into above equation

y = 7(1/14)4 – 2(1/14)3 – 4(1/14)2 – 5(1/14) + 4

y = 18.2 – 0.0003 – 0.005 – 0.35 + 4

y = 21.8

Therefore maximum value of a function is 21.8

 

Maximum Value of Function – Practice Problems

 

Problem 1: Find the maximum value of a function; y = 5x2 + 2x + 3.

Answer: 2.8

Problem 2: Find the maximum value of a function; y = 4x2 + 16x + 2.

Answer: 50