Question

The absolute maximum value of the function `f(x) = 4x - 1/2 x^2` in the interval `[-2, 9/2]` is ______.

Options

• 8

• 9

• 6

• 10

Answer 1

The absolute maximum value of the function `f(x) = 4x - 1/2 x^2` in the interval `[-2, 9/2]` is 8.

Explanation:

Given, `f(x) = 4x - 1/2 x^2`

∴ `f^'(x) = 4 - 1/2 (2x) = 4 - x`

Put f'(x) = 0

`\implies` 4 – x = 0

`\implies` x = 4

Then, we evaluate the f at critical point x = 4 and at the end points of the interval `[-2, 9/2]`.

`f(4) = 16 - 1/2 (16)`

= 16 – 8

= 8

`f(-2) = -8 - 1/2 (4)`

= –8 – 2

= –10

`f(9/2) = 4(9/2) - 1/2(9/2)^2`

= `18 - 81/8`

= 7.875

Thus, the absolute maximum value of f on `[-2, 9/2]` is 8 occurring at x = 4.