List of top Mathematics Questions on Differentiation asked in IIT JAM MA

Let the function \( f:\mathbb{R}^2\to \mathbb{R} \) be defined by

Let \( z=\cos(4x+5y) \), where \( x=\frac{\pi}{2}+2\theta \), \( y=-\left(\frac{\pi}{4}+\theta\right) \).

Let \[ f(x, y) = \begin{cases} \frac{x^2 y}{1 + x^2} \sin\left(\frac{1}{x}\right), & \text{if} \ x \neq 0 \\ 0, & \text{if} \ x = 0 \end{cases} \]
Which of the following statements is/are TRUE?

Which of the following functions \( f: \mathbb{R} \to \mathbb{R} \) has/have a local minimum at \( x = 0 \)?

Which of the following functions is/are differentiable at \( x = 1 \)?

If \[ y(x) = \int_{\sqrt{x}}^x e^t \, dt, \quad x>0 \] then \( y'(1) = \) .............

Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(2) = 2 \) and \[ |f(x) - f(y)| \leq 5| |x - y|^{3/2}. \] For all \( x \in \mathbb{R}, y \in \mathbb{R} \), let \( g(x) = x^3 f(x) \). Then \[ g'(2) = \]