Consider the function

$$ f(\mathrm{x})=\frac{x^3}{3}+\frac{7}{2} x^2+10 x+\frac{133}{2}, x \in[-8,0] . $$

Which of the following statements is/are correct?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice-differentiable function and suppose its second derivative

satisfies $f^{\prime \prime}(x)>0$ for all $x \in \mathbb{R}$. Which of the following statements is/are ALWAYS correct?

An $n \times n$ matrix $A$ with real entries satisfies the property: $\|A x\|^2=\|x\|^2$ for all $x \in R^n$ where $\|\cdot\|$ denotes the Euclidean norm. Which of the following statements is/are ALWAYS correct?

Consider a coin-toss experiment where the probability of head showing up is $p$. In the $i^{\text {th }}$ coin toss, let $X_i=1$ if head appears, and $X_i=0$ if tail appears.

Consider

$$ \hat{p}=\frac{1}{n} \sum_{i=1}^n X_i $$

where $n$ is the total number of independent coin tosses.