Question

Let $u: \mathbb{R} \to \mathbb{R}$ be a twice continuously differentiable function such that $u(0) > 0$ and $u'(0) > 0$. Suppose $u$ satisfies $u''(x) = \frac{u(x)}{1 + x^2}$
for all $x \in \mathbb{R}$.
Consider the following two statements:
I. The function $uu'$ is monotonically increasing on $[0, \infty)$.
II. The function $u$ is monotonically increasing on $[0, \infty)$.
Then which one of the following is correct?

• A. Both I and II are false.
• B. Both I and II are true.
• C. I is false, but II is true.
• D. I is true, but II is false.

Answer 1

Answer: (B)

Both I and II are true.

Answer 2

The correct option is (B): Both I and II are true.