GATE 2024 ST – Question 36
For f(x,y)=108xy−2x²y−2xy², which is not true?
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Correct answer: (B) A local minimum at (0,0)
Explanation
Stationarity is y(54−2x−y)=0,x(54−x−2y)=0, giving (0,0),(54,0),(0,54),(18,18). The Hessian is [[−4y,108−4x−4y],[108−4x−4y,−4x]]. Its determinant is negative at the first three points (saddles), while at (18,18) it is positive with negative first diagonal entry (local maximum). The origin is a saddle, not a local minimum. **B**.