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
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.
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