Part CCSIR NET June 2025bounded-plus-convex-forces-f-prime-to-0-fast-enough-that-x-f-prime-goes-too

Bounded plus convex forces f prime to 0 fast enough that x f prime goes too

Let f be a bounded, twice continuously differentiable real-valued function on $(0, \infty)$ such that f″(x) ≥ 0 for all $x \in (0, \infty)$ . Which of the following statements are true?

A.f′(x) ≤ 0 for all x > 0.✓
B. $l i m_{x \to \infty}$ f′(x) = 0.✓
C. $l i m_{x \to \infty}$ x f′(x) need not exist.
D. $l i m_{x \to \infty}$ x f′(x) = 0.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

Drill statements like this

Related counterexample: Differentiable ⇒ continuously differentiable

Bounded plus convex forces f prime to 0 fast enough that x f prime goes too

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