Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of x at which the function is maximum or minimum. y = 2x² - 4x + 7

Solution:

Given, the quadratic function is y = 2x² - 4x + 7

We have to find the value of x at which the function is maximum or minimum.

y = 2(x² - 2x + 7/2)

By completing the squares,

x² - 2x + 7/2 = x² - 2x + 1 - 1 + 7/2

= (x - 1)² - 1 + 7/2

So, y = 2((x - 1)² - 1 + 7/2)

y = 2[(x - 1)² + (-2+7)/2]

y = 2[(x - 1)² + 5/2]

y = 2(x - 1)² + 5

To find the minimum value,

2(x - 1)² = 0

(x - 1)² = 0

Taking square root,

x - 1 = 0

x = 1

Therefore, x = 1 is the minimum point.

On differentiating,

y = 2x² - 4x + 7

y’ = 4x - 4

To find the maximum value,

4x - 4 = 0

4(x - 1) = 0

x - 1 = 0

x = 1

Therefore, the value of x at which the function is maximum or minimum is x = 1.

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Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of x at which the function is maximum or minimum. y = 2x² - 4x + 7

Summary:

The quadratic function y = 2x²- 4x + 7 by completing the squares is y = 2(x - 1)² + 5. The value of x at which the function is maximum or minimum is x = 1.