PEA 2024 PEA Q6 | ISI MSQE

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2024 PEA Q6

Let $f:\mathbb{R}\to\mathbb{R}$ be the function $f(x)=|x|+x^{2}\ \forall x\in\mathbb{R}$ . Which of the following statements about $f$ is correct?

Reveal answer and solution

Answer

Solution

1
At $x=0$ , the left derivative of $|x|$ is $-1$ and the right derivative is $+1$ ; hence $|x|$ is not differentiable at $0$ , and so $f$ is not differentiable at $0$ . However $|x|$ and $x^{2}$ are both convex on $\mathbb{R}$ , and the sum of convex functions is convex. Hence $f$ is convex but not differentiable.

Answer structure / marking notes

A non-differentiable function can still be convex (e.g., $|x|$ ).

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Content note

Imported from public/resources/isi/msqe/solutions/pea/2024/ISI_MSQE_PEA_2024_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.