If x + y = 8, then the maximum value of xy is :A. 8 B. 16 C. 20 D.

1.	If x + y = 8, then the maximum value of xy is :A. 8 B. 16 C. 20 D. 24
Answer» Option : (B) x + y = 8 ⇒ y = 8 - x xy = x(8 - x) Let f(x) = 8x - x² Differentiating f(x) with respect to x, we get f’(x) = 8 - 2x Differentiating f’(x) with respect to x, we get f’’(x) = -2 < 0 For maxima at x = c, f’(c) = 0 and f’’(c) < 0 f’(x) = 0 ⇒ x = 4 Also, f’’(4) = -2 < 0 Hence, x = 4 is a point of maxima for f(x) and f(4) = 16 is the maximum value of f(x).

If x + y = 8, then the maximum value of xy is :A. 8 B. 16 C. 20 D. 24

Answer»

Option : (B)

x + y = 8

⇒ y = 8 - x

xy = x(8 - x)

Let f(x) = 8x - x²

Differentiating f(x) with respect to x, we get

f’(x) = 8 - 2x

Differentiating f’(x) with respect to x, we get

f’’(x) = -2 < 0

For maxima at x = c,

f’(c) = 0 and f’’(c) < 0

f’(x) = 0 ⇒ x = 4

Also,

f’’(4) = -2 < 0

Hence,

x = 4 is a point of maxima for f(x) and

f(4) = 16 is the maximum value of f(x).