Math, asked by sravani2215, 3 months ago

If a is real, then the Range of 2x^ 2 -6x+8 is ​

Answers

Answered by amitnrw
7

Range of  2x² - 6x + 8  is [7/2,  ∞) if x is real

Given:

  • x is real

To Find :

  • Range of 2x² - 6x + 8

Assume that f(x) = 2x² - 6x + 8

Concept to be used :

To find Minimum/Maximum value derivative of f(x) to be found.

f(x) = 2x² - 6x + 8

Step 1 :

Taking derivative

f'(x) = 4x - 6

Step 2 :

Equate derivative with 0

=> 4x - 6 = 0

Step 3 :

Solve for x

=> x  = 6/4  = 3/2

Step 4 :

Find 2nd derivative

f''(x) = 4  > 0

Hence f(x) minimum  value at  x = 3/2

Step 5 :

Calculate f(x) at x = 3/2

= 2(3/2)² - 6(3/2)  + 8

= 4.5  - 9  + 8

= 3.5

= 7/2

Minimum value is 7/2  and no maximum value

Hence Range of  2x² - 6x + 8  is [7/2,  ∞)

(There was a typing mistake in the Question , probably a is real should be x is real)

Another method

2x² - 6x + 8

Step 1 :

Taking 2 as common  factor

= 2(x²  - 3x  + 4)

Step 2:

Completing the square

= 2((x - 3/2)²  - 9/4 + 4)

= 2(x - 3/2)² + 7/2

As (x - 3/2)² can not be negative Hence minimum value = 7/2

And there is not limit for maximum value.

Hence Range of  2x² - 6x + 8  is [7/2,  ∞)

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