GATE 2022 MA – Question 32

Calculus · Multivariable Differentiation and Optimization · 1 mark · Numerical answer

For x,y≠0, f(x,y)=x²+xy+y²+1/x+1/y has a local minimum at (a,b). Find a³+b³ to two decimals.

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Correct answer: 0.65 to 0.68

Explanation

Stationarity gives 2a+b=1/a² and a+2b=1/b². Their sum implies a+b>0. Subtracting gives (a−b)[1+(a+b)/(a²b²)]=0, so a=b. Then 3a³=1. The Hessian there has diagonal 2+2/a³=8 and off-diagonal 1, with positive eigenvalues, confirming a minimum. Thus a³+b³=2/3=**0.67**.

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