The sum of the eigenvalues of the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2$ is ______ (rounded off to the nearest integer).

Consider a vector $\vec{u} = 2\hat{x} + \hat{y} + 2\hat{z}$, where $\hat{x}$, $\hat{y}$, $\hat{z}$ represent unit vectors along the coordinate axes $x$, $y$, $z$ respectively. The directional derivative of the function $f(x, y, z) = 2\ln(xy) + \ln(yz) + 3\ln(xz)$ at the point $(x, y, z) = (1, 1, 1)$ in the direction of $\vec{u}$ is

Let $f(t)$ be a real-valued function whose second derivative is positive for $- \infty < t < \infty$. Which of the following statements is/are always true?

Consider the function $f(t) = (\text{max}(0,t))^2$ for $- \infty < t < \infty$, where $\text{max}(a,b)$ denotes the maximum of $a$ and $b$. Which of the following statements is/are true?