find the maximum and minimum value of fuction
Answers
Answer:
The turning points of a graph
WE SAY THAT A FUNCTION f(x) has a relative maximum value at x = a, ...
We say that a function f(x) has a relative minimum value at x = b, ...
The value of the function, the value of y, at either a maximum or a minimum is called an extreme value.
f '(x) = 0.
Answer:
1. Differentiate the given function.
2. let f'(x) = 0 and find critical numbers
3. Then find the second derivative f''(x).
4. Apply those critical numbers in the second derivative.
5. The function f (x) is maximum when f''(x) < 0
6. The function f (x) is minimum when f''(x) > 0
7. To find the maximum and minimum value we need to apply those x values in the original function.
Step-by-step explanation:
Examples:
1.
Determine maximum values of the functions
y = 4x - x2 + 3
Solution :
f(x) = y = 4x - x2 + 3
First let us find the first derivative
f'(x) = 4(1) - 2x + 0
f'(x) = 4 - 2x
Let f'(x) = 0
4 - 2x = 0
2 (2 - x) = 0
2 - x = 0
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 = 4x - x2 + 3
The given function is the equation of parabola.
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.
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