List of top Calculus Questions on Functions of two or more variables

The maximum value of \( f(x, y) = 49 - x^2 - y^2 \) on the line \( x + 3y = 10 \) is __________.

If the function \( f(x, y) = x^2 + xy + y^2 + \frac{1}{x} + \frac{1}{y} \), \( x \neq 0, y \neq 0 \) attains its local minimum value at the point \( (a, b) \), then the value of \( a^3 + b^3 \) is __________ (round off to TWO decimal places).

The rate of change of \( f(x, y, z) = x + x \cos z - y \sin z + y \) at \( P_0(2, -1, 0) \) in the direction from \( P_0(2, -1, 0) \) to \( P_1(0, 1, 2) \) is __________.

Number of finite non-isomorphic groups with exactly 3 conjugacy classes_____.