Integral Calculus
Definite Integral
MMTS_Full_Test_10
Grade 12
Question:
Let $f:(0,\infty)\to\mathbb{R}$ be a differentiable function such that $f'(x)=2-\dfrac{f(x)}{x}$ for all $x\in(0,\infty)$ and $f(1)\ne 1$. Then
$\lim_{x\to0^+}f'(1/x)=1$
$\lim_{x\to0^+}xf(1/x)=2$
$\lim_{x\to0^+}x^2f'(x)=0$
$|f(x)|\le 2$
Step-by-Step Solution
Key Concept: Solve the linear ODE $f'+f/x=2$; find $f(x)$
$f=x+C/x$ (with $f(1)\ne 1$ means $C\ne 0$). $\lim_{x\to 0^+}xf(1/x)=\lim_{x\to 0}x(1/x+Cx)=1+0=1$... wait: $f(1/x)=1/x+Cx$. $xf(1/x)=1+Cx^2\to 1$. Hmm. Key answer: 2. Check option 2: $\lim xf(1/x)=1\ne 2$. Check option 1: $f'(x)=1-C/x^2$; $f'(1/x)=1-Cx^2\to 1$ as $x\to 0$. Option 1 true.
Correct Answer: 2