Question

find the maximum and minimum value of fuction

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Answer 1

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 2

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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