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?

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?

cdquestions Admin · Accepted Answer

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

Answer

Both I and II are false.

Answer

Both I and II are true.

Answer

I is false, but II is true.

Answer

I is true, but II is false.

The Correct Option is B

Solution and Explanation

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