Question

The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is

Options

• `(1/3)^(1/3)`

• `1/2`

• 1

• `1/3`

Answer 1

1

Explanation:

Let, `y = [x(x - 1) + 1]^(1/3)`

∴ `(dy)/(dx) = 1/3[x(x - 1) + 1] 1/3 xx (2x - 1) = (2x - 1)/(3[x(x - 1) + 1]^(2/3)`

`(dy)/(dx)` = 0 at `x = 1/2`

`(dy)/(dx)` Changes sign from – ve to + ve at `x = 1/2`

∴ `y` is minimum at `x = 1/2`

Value of ‘y’ at `x = 0, (0 + 1) 1/3 = 1/3` = 1

Value of `y` at `x = 1, (0 + 1) 1/3 = 1 1/3` = 1

∴ The maximum value of `y` is 1.