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GATE 2021 MA – Question 49

Calculus · Multivariable Differentiation and Optimization · 2 marks · Multiple choice

For f(x,y)=4xy−2x²−y⁴, which description holds?

  1. There is a local maximum point and a saddle point.
  2. There is a local minimum point and a saddle point.
  3. There is a local maximum point and a local minimum point.
  4. There are two saddle points.

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Correct answer: (A) There is a local maximum point and a saddle point.

Explanation

Stationarity gives x=y and y−y³=0, so critical points are (0,0),(1,1),(−1,−1). The Hessian has entries (−4,4;4,−12y²) and determinant 48y²−16. It is indefinite at zero (saddle), and negative definite at the other two points (local maxima). Thus **A**; its wording asserts existence, not uniqueness of the maximum point.

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