Question

If x + y = 8, then the maximum value of x²y is ______.

Options

• `2048/9`

• `2048/3`

• `2048/81`

• `2048/27`

Answer 1

If x + y = 8, then the maximum value of x²y is `underlinebb(2048/27)`.

Explanation:

x + y = 8   ...(1)

Let f(x) = x²y = x²(8 – x)   ...[From (1)]

∴ f(x) = 8x² – x³

∴ f'(x) = 16x – 3x²

and f″(x) = 16 – 6x

Put f′(x) = 0,

∴ 16x – 3x² = 0

∴ x(16 – 3x) = 0

∴ x = 0, x = `16/3`

Now `f^('')(x)|_(x = 0)` = 16 > 0

and `f^('')(x)|_(x = 16/3) = 16 - 6 xx 16/3`

= 16 – 32

= – 16 < 0

f(x) is maximum at x = `16/3`

Max. f(x) = `8(256/9) - 16/3(256/9)`

= `256/9(8 - 16/3)`

= `256/9(8/3)`

= `2048/27`