The GATE Grind

GATE Calculus: Multivariable Differentiation and Optimization – Previous Year Questions

18 GATE previous year questions on Multivariable Differentiation and Optimization (Calculus, Mathematics) with answers and explanations, from every paper.

  1. GATE 2021 MA Q25 (1 mark, Numerical answer) – Differentiable f:ℝ²→ℝ has directional derivatives at zero D u f=√5 and D v f=√2, where u=(1/√5,2/√5),v=(1/√2,−1/√2). Find f x(0,0)+f y(0,0).
  2. GATE 2021 MA Q47 (2 marks, Multiple choice) – Define f(x,y)=√(x²+y²) sin(y²/x) for x≠0 and zero for x=0. P:f is continuous but not differentiable at (0,0). Q:every unit-direction derivative exists…
  3. GATE 2021 MA Q49 (2 marks, Multiple choice) – For f(x,y)=4xy−2x²−y⁴, which description holds?
  4. GATE 2022 MA Q31 (1 mark, Numerical answer) – Find the maximum of 49−x²−y² on x+3y=10.
  5. GATE 2022 MA Q32 (1 mark, Numerical answer) – For x,y≠0, f(x,y)=x²+xy+y²+1/x+1/y has a local minimum at (a,b). Find a³+b³ to two decimals.
  6. GATE 2022 MA Q60 (2 marks, Numerical answer) – For f(x,y,z)=x+x cos z−y sin z+y, find the rate of change at P₀=(2,−1,0) toward P₁=(0,1,2).
  7. GATE 2023 MA Q11 (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.
  8. GATE 2023 MA Q52 (2 marks, Multiple select) – Let f(x,y)=2x²y/(x²+y²) away from the origin and f(0,0)=0. Which statements hold?
  9. GATE 2024 MA Q26 (1 mark, Numerical answer) – f:ℝ⁴→ℝ is differentiable and homogeneous of degree 3, with f(p)=6 at p=(1,1/2,1/3,1/4). Find 12f x₁(p)+6f x₂(p)+4f x₃(p)+3f x₄(p) (integer).
  10. GATE 2024 MA Q34 (1 mark, Numerical answer) – For positive x,y,z with xy+yz+zx=12, find maximum xyz to two decimals.
  11. GATE 2024 MA Q39 (2 marks, Multiple choice) – On M₂(ℝ)≅ℝ⁴, g(A)=trace(A²). Classify the zero matrix.
  12. GATE 2024 MA Q54 (2 marks, Multiple select) – f(x,y)=(1−cos(x²/y²))√(x²+y²) for y≠0, and zero otherwise. Which statements hold at (0,0)?
  13. GATE 2024 MA Q64 (2 marks, Numerical answer) – Find the global maximum of (x²+y²)e(2−x−y) for x,y≥0, to two decimals.
  14. GATE 2025 MA Q18 (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?
  15. GATE 2025 MA Q26 (1 mark, Multiple select) – For (x,y)≠(0,0), f=(x2-y2)/(x2+y2)+x[1/(x2+y2)]. S1: lim (x→0) lim (y→0) f exists; S2: lim (y→0) lim (x→0) f exists; S3: joint limit at origin exists.…
  16. GATE 2025 MA Q29 (1 mark, Numerical answer) – At P=(1,1,√2), a unit vector a is tangent to the intersection 2x²+3y²−z²=3 and x²+y²=z². For f=x²+2y²−2√11 z, find the absolute directional derivative…
  17. GATE 2025 MA Q39 (2 marks, Multiple choice) – C is the intersection z²=x²+y² and 4x+z=7. A point P on C has minimum distance to the xy-plane. Find its distance from the origin.
  18. GATE 2026 MA Q22 (1 mark, Multiple select) – f(x,y)=xx2+y2/ x for x ≠ 0 and 0 for x = 0. Which statements hold at (0,0)?