Find the Maximum and Minimum Values of the Function Examples :
In this section, we will see some example problems of finding maximum and minimum values of the function.
The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function.
Question 1 :
Find the maximum and minimum value of the function
2x3 - 15x2 + 36x + 18
Solution :
Let y = f(x) = 2x3 - 15x2 + 36x + 18
f'(x) = 2(3x2) - 15 (2x) + 36 (1) + 0
f'(x) = 6x² - 30x + 36
f'(x) = 0
6x² - 30x + 36 = 0
÷ by 6 => x² - 5 x + 6 = 0
x - 2 = 0 x = 2 |
x - 3 = 0 x = 3 |
f'(x) = 6x² - 30x + 36
f''(x) = 6(2x) - 30 (1) + 0
f''(x) = 12 x - 30
Put x = 2
f''(2) = 12(2) - 30
= 24 - 30
f''(2) = -6 < 0 Maximum
To find the maximum value let us apply x = 2 in the given function.
f(2) = 2 (2)³ - 15 (2)² + 36 (2) + 18
= 2(8) - 15(4) + 72 + 18
= 16 - 60 + 72 + 18
= 106 - 60
f(2) = 46
Put x = 3
f''(3) = 12(3) - 30
= 36 - 30
f''(3) = 6 > 0 Minimum
To find the minimum value let us apply x = 3 in the given function.
f (3) = 2 (3)³ - 15 (3)² + 36 (3) + 18
= 2(27) - 15(9) + 108 + 18
= 54 - 135 + 108 + 18
= 180 - 135
= 45
Therefore the maximum value is 46 and minimum value is 45.
Question 2 :
Find the maximum and minimum value of the function x3 - 6 x2 + 9 x + 1.
Solution :
Let y = f (x) = x3 - 6 x2 + 9 x + 1
f'(x) = 3x² - 6 (2x) + 9 (1) + 0
f'(x) = 3x² - 12x + 9
f'(x) = 0
3x² - 12x + 9 = 0
÷ by 3 => x² - 4 x + 3 = 0
x - 1 = 0 x = 1 |
x - 3 = 0 x = 3 |
f'(x) = 3x² - 12x + 9
f''(x) = 3 (2 x) - 12 (1) + 0
f''(x) = 6 x - 12
Put x = 1
f''(1) = 6(1) - 12
= 6 - 12
f''(1) = -6 < 0 Maximum
To find the maximum value let us apply x = 1 in the original function
f (x) = x³ - 6 x² + 9 x + 1
f (1) = (1)³ - 6 (1)² + 9 (1) + 1
= 1 - 6(1) + 9 + 1
= 1 - 6 + 10
= 11 - 6
= 5
Put x = 3
f''(3) = 6(3) - 12
= 18 - 12
f '' (3) = 6 > 0 Minimum
To find the minimum value let us apply x = 3 in the original function
f(x) = x3 - 6x2 + 9x + 1
f (3) = 33 - 6 (3)2 + 9 (3) + 1
= 27 - 6(9) + 27 + 1
= 54 + 1 - 54
= 1
Therefore the maximum value is 5 and minimum value is 1.
After having gone through the stuff given above, we hope that the students would have understood how to find maximum and minimum value of the function.
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