For f(x,y)=5-(+)x2+ y2+(+1)y3+x3, suppose (4,0) and (0,−1/2) are critical points. Which holds?

GATE 2020 MA – Question 19

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

For $$f(x,y)=5-(\alpha+\beta)x^2+\beta y^2+(\alpha+1)y^3+x^3,$$ suppose (4,0) and (0,−1/2) are critical points. Which holds?

$(4,-1/2)$ is a local maximum.
$(4,-1/2)$ is a saddle point.
$\alpha=4,\beta=2$
$(4,-1/2)$ is a local minimum.

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Correct answer: (B) $(4,-1/2)$ is a saddle point.

Explanation

From $f_x(4,0)=0$ obtain α + β = 6. From $f_y(0,-1/2)=0$ obtain $\beta=3(\alpha+1)/4$. These give α = β = 3. At (4,−1/2), the mixed derivative is zero, while $f_{xx}=12$ and $f_{yy}=-6$. The Hessian has opposite-sign eigenvalues and negative determinant, so this critical point is a saddle. Select B.

GATE 2020 MA – Question 19

Explanation

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