GATE 2024 MA – Question 64

Calculus · Multivariable Differentiation and Optimization · 2 marks · Numerical answer

Find the global maximum of (x²+y²)e^(2−x−y) for x,y≥0, to two decimals.

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Correct answer: 3.99 to 4.01

Explanation

Let s=x+y≥0. Since x²+y²≤s², the expression is at most s²e^(2−s). Its derivative is e^(2−s)s(2−s), so its maximum occurs at s=2 and equals 4. Equality in x²+y²≤s² requires xy=0, attained at (2,0) or (0,2). **4.00**.

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