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

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Maximum and Minimum Examples to Monotonic Function