Question

If log₁₀ (x³ + y³) – log₁₀ (x² + y² – xy) ≤ 2, then the maximum value of xy for all x ≥ 0, y ≥ 0 is

(a) 1200 

(b) 2500 

(c) 3000 

(d) 3500

Answer 1

(b) 2500

log₁₀ (x³ + y³) – log₁₀ (x² + y² – xy) < 2

⇒ (x + y) < 10² ⇒ x + y < 100 

Now x > 0, y > 0 ⇒ AM > GM

∴ \(\frac{x+y}{2}≥\sqrt{xy}\)

⇒ \(\frac{100}{2}≥\frac{x+y}{2}≥\sqrt{xy}\)

⇒ \(\sqrt{xy}≤50\) ⇒ xy ≤ 2500.

Maximum value of xy = 2500