GATE 2025 MA – Question 18

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

g(x,y)=f(x,y)e^(2x+3y), where f is a nonzero C¹ homogeneous function of degree four on ℝ². On which listed set does xg_x+yg_y=0 hold?

All ℝ²
All points on 2x+3y+4=0
Every point except on 2x+3y+4=0
All points on 2x+3y=0

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Correct answer: (B) All points on 2x+3y+4=0

Explanation

Euler’s identity gives xf_x+yf_y=4f. Thus $xg_x+yg_y=e^{2x+3y}(4+2x+3y)f$. This vanishes everywhere on **2x+3y+4=0, B**. It may also vanish at zeros of f, but that does not change the guaranteed listed line.

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