Let $f(.)$ be a twice differentiable function from $ \mathbb{R}^{2} \rightarrow \mathbb{R}$. If $P, \mathbf{x}_{0} \in \mathbb{R}^{2}$ where $\vert \vert P\vert \vert$ is sufficiently small (here $\vert \vert . \vert \vert$ is the Euclidean norm or distance function), then $f (\mathbf{x}_{0} + p) = f(\mathbf{x}_{0}) + \nabla f(\mathbf{x}_{0})^{T}p + \dfrac{1}{2} p^{T} \nabla^{2}f(\psi)p$ where $\psi \in \mathbb{R}^{2}$ is a point on the line segment joining $\mathbf{x}_{0}$ and $\mathbf{x}_{0} + p$. If $\mathbf{x}_{0}$ is a strict local minimum of $f (\mathbf{x})$, then which one of the following statements is TRUE?

The figure shows the plot of a function over the interval [-4, 4]. Which one of the options given CORRECTLY identifies the function?

Given $\int^{\infty}_{-\infty}e^{-x^2}dx=\sqrt{\pi}$

If a and b are positive integers, the value of $\int^{\infty}_{-\infty}e^{-a(x+b)^2}dx$ is _________.

The limit

$\rm p = \displaystyle\lim_{x \rightarrow \pi} \left( \frac{x^2 + α x + 2 \pi^2}{x - \pi + 2 \sin x} \right)$

has a finite value for a real α. The value of α and the corresponding limit p are

In the closed interval $[0,3]$, the minimum value of the function $f$ given below is $f(x)=2 x^3-9 x^2+12 x$

If the value of the double integral

$\int_{x=3}^{4} \int_{y=1}^{2} \frac{dydx}{(x + y)^2}$

is $\log_e(\frac{a}{24})$, then $a$ is __________ (answer in integer).