Maximum and minimum

Maximum and Minimum

In this page maximum and minimum we are going to see how to find maximum-and-minimum values for the given function and example problems.

Definition:

The y co ordinate of the turning point at which the function changes from increasing to decreasing is called maximum values of the function.

Procedure for finding maximum-and-minimum values of any function

Determine the first derivative
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 original function.

Now let us see some examples to understand this topic.

Example 1:

Determine maximum values of the functions y = 4 x - x² + 3

Solution:

y = f (x) = 4 x - x² + 3

first let us find the first derivative

f ' (x ) = 4 (1) - 2 x + 0

f ' (x) = 4 - 2 x

Let f ' (x) = 0

4 - 2 x = 0

2 (2 - x) = 0

2 - x = 0

- x = -2

x = 2

Now let us find the second derivative

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

f ''(x) = -2 < 0 Maximum

To find the maximum value we have to apply x = 2 in the original function

f (2) = 4 (2) - 2² + 3

f (2) = 8 - 4 + 3

f (2) = 11 - 4

f (2) = 7

therefore the maximum value is 7 at x = 2. Now let us check this in the graph.

Checking:

y = 4 x - x² + 3

y = - x² + 4 x + 3

y = - (x² - 4 x - 3)

y = - (x² - 2 (x) (2) + 2² - 2² - 3)

y = - [(x - 2)² - 4 - 3]

y = - [(x - 2)² - 7]

y = - (x - 2)² + 7

y - 7 = -(x - 2)²

( y - k ) = -4a ( x - h )²

Here (h,k) is (2,7) and the parabola is open downward

From the figure above we come to know that the maximum value is 7 at x = 2. Like this you can find more and more examples in the following link.You can also find practical problems with brief solution.