List of top Engineering Mathematics Questions on Functions of Two Variables asked in GATE XE

Consider the function \[ f(x,y) = x^2y + 2xy^2 - 2x^2y^2. \] Then which one of the following statements is correct?
The surface area of the portion of the paraboloid \[ z=x^2+y^2 \] that lies between the planes $z=0$ and $z=\tfrac{1}{4}$ is

Let \( f : (0, \infty) \to \mathbb{R} \) be the continuous function such that:

\[ f(x) = 2 + \frac{g(x)}{x} \quad \text{for all} \ x > 0, \quad g(x) = \int_1^x f(t) \, dt \quad \text{for all} \ x > 0. \]

Then \( f(2) \) is equal to:

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x, y) = 4xy - 2x^2 - y^4 + 1. \] The number of critical points where \( f \) has local maximum is equal to ________.