GATE 2020 ST – Question 36

Calculus · Multivariable Calculus · 2 marks · Multiple choice

For f(x,y)=x⁴−2x³y+16y+17 on R², which is true?

Local minimum at(2,4/3)
Local maximum at(2,4/3)
Saddle point at(2,4/3)
f is bounded

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Correct answer: (C) Saddle point at(2,4/3)

Explanation

Gradient equations give fy=−2x³+16=0→x=2, then fx=4x³−6x²y=0→y=4/3. At this point fxx=16,fxy=−24,fyy=0. Hessian determinant=16(0)−(−24)²=−576<0, so it is a saddle, C.

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