GATE 2023 MA – Question 11

Calculus · Multivariable Differentiation and Optimization · 1 mark · Multiple choice

For f(x,y)=x²−(3/2)xy² and g(x,y)=4x⁴−5x²y+y² on ℝ², P:f has a saddle at (0,0); Q:g has a saddle there. Choose truth values.

Both P and Q true
P false, Q true
P true, Q false
Both false

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Correct answer: (A) Both P and Q true

Explanation

Both vanish and have zero gradient at the origin. Along y=0, f=x²>0, while along x=3y²/4 it is −9y⁴/16<0. Factor g=(y−x²)(y−4x²): it is positive along y=0 and negative along y=2x². Both change sign arbitrarily near the critical point. **A**.

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