Check $x=0$: $f(x)=|x-\pi|(e^{|x|}-1)\sin|x|$. Near $x=0$, both $(e^{|x|}-1)$ and $\sin|x|$ have factors of $|x|$, so $f(x)\approx |x-\pi|\cdot|x|^2$ near 0, which is differentiable at 0.

Check $x=\pi$: near $\pi$, $(e^{|x|}-1)\sin|x|$ is smooth and non-zero ($e^\pi\sin\pi=0$!). Since $\sin\pi=0$, the product $(e^{|x|}-1)\sin|x|$ vanishes at $\pi$, making $f(x)=0$ near $\pi$ — wait, $\sin\pi=0$, so $f(\pi)=0$. Checking differentiability more carefully shows $f$ is differentiable at $\pi$ too.

Therefore $S=\boldsymbol{\emptyset}$.

A private limited company offers limited liability, protecting Jamie's personal assets. As a separate legal entity, it allows him to be the sole owner (shareholder) and receive a salary as a director. A sole trader or partnership would have unlimited liability, risking personal assets.

Autonomic (sympathetic/parasympathetic) controls involuntary functions: heart rate, digestion, salivation, smooth muscle. Thoughts and emotions are controlled by the central nervous system (cerebrum/limbic system).

In formal British English, shall is used with the first person (I/we) to express simple future. Will with first person traditionally expresses determination or a promise rather than plain future. However, in modern usage both are acceptable. In the context of this MCQ (a Pakistani exam following traditional grammar rules), shall is the expected answer for first-person future.

Correct Answer: B — shall.

• \(r = -0.82\) indicates a strong negative correlation.
• A negative sign means the two variables move in opposite directions: as patient anxiety increases, recovery time decreases — or vice versa.
• The magnitude \(|r| = 0.82\) is close to 1, indicating a strong relationship.