FIND THE MAXIMUM AND MINIMUM VALUES OF THE FUNCTION EXAMPLES

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.

• Differentiate the given function.
• let f'(x)  =  0 and find critical numbers
• Then find the second derivative f''(x).
• Apply those critical numbers in the second derivative.
• The function f (x) is maximum when f''(x) < 0
• The function f (x) is minimum when f''(x) > 0
• To find the maximum and minimum value we need to apply those x values in the given function.

Finding the Maximum and Minimum Values of the Function Examples

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  =  0x  =  2 x - 3  =  0x  =  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  =  0x  =  1 x - 3  =  0x  =  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.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6