Limits, Continuity & Differentiability
Limit Condition Giving ODE — Function Value
nta_pyq_2024_apr
Grade None
Question:
Let $f$ be a differentiable function in the interval $(0,\infty)$ such that $f(1)=1$ and $\displaystyle\lim_{t\to x}\frac{t^2f(x)-x^2f(t)}{t-x}=1$ for each $x>0$. Then $2f(2)+3f(3)$ is equal to
Step-by-Step Solution
Key Concept: The limit equals $2xf(x)-x^2f'(x)=1$ (via L'Hôpital w.r.t. $t$). This is a linear ODE: $\frac{dy}{dx}-\frac{2}{x}y=-\frac{1}{x^2}$.
$f(x)=\frac{1}{3x}+\frac{2x^2}{3}$. $2f(2)+3f(3)=24$.
Correct Answer: 24