## Maximum and Minimum Examples

In this page maximum and minimum examples we are going to see some more example to understand the topic maximum and minimum.

Example :

Find the maximum and minimum value of the function 2 x³ + 3 x² - 36 x + 1

Solution:

Let y = f (x) = 2 x³ + 3 x² - 36 x + 1

f ' (x) = 2(3x²) + 3 (2x) - 36 (1) + 0

f ' (x) = 6x² + 6x - 36

set f ' (x) = 0

6x² + 6x - 36 = 0

÷ by 6 => x² + x - 6

(x - 2) (x + 3) = 0

x - 2 = 0        x + 3 = 0

x = 2              x = - 3

f ' (x) = 6x² + 6x - 36

f '' (x) = 6 (2x) + 6(1) - 0

f '' (x) = 12x + 6

Put  x = 2

f '' (2) = 12(2) + 6

= 24 + 6

f '' (2) = 30 >0 Minimum

To find the minimum value let us apply x = 2 in the original function

f (2) = 2 (2)³ + 3 (2)² - 36 (2) + 1

= 2(8) + 3(4) - 72 + 1

= 16 + 12 - 72 + 1

= 29 - 72

= -43

Put  x = -3

f '' (-3) = 12(-3) + 6

= -36 + 6

f '' (-3) = -30 >0 Maximum

To find the maximum value let us apply x = -3 in the original function

f (-3) = 2 (-3)³ + 3 (-3)² - 36 (-3) + 1

= 2(-27) + 3(9) + 108 + 1

= -54 + 27 + 109

= -54 + 136

= 82

Therefore the minimum value = -43 and

The maximum value = 82

maximum and minimum examples

 Questions Solution

 (1) Find the maximum and minimum value of the function 2 x³ - 15 x² + 36 x + 18(2) Find the maximum and minimum value of the function x³ - 6 x² + 9 x + 1(3) Find the maximum and minimum value of the function 2 x³ - 3 x² - 12 x + 5(4) Find the maximum and minimum value of the function x³ - 3 x² - 9 x + 12(5) Find the maximum and minimum value of the function 4 x³ - 18 x² + 24 x - 7 SolutionSolutionSolutionSolutionSolution

Maximum and Minimum Examples to Monotonic Function