An ODE in Disguise

★ 7/10functions Extrema calculus Substitution Monotonicity

Problem

A function $f(x)$ on $(0, +\infty)$ has derivative $f'(x)$ satisfying

x^2f'(x) + xf(x) = \mathrm{e}^{x/2}, \qquad f(2) = \frac{\mathrm{e}}{2}.

Determine, with proof, which of the following are true:

1. $f(x)$ has a local minimum; 2. $f(x)$ is increasing on its domain; 3. $H(x) = xf(x) - \mathrm{e}\ln x$ is decreasing on $(0, 2)$ ; 4. the solution set of $f(x) > \dfrac{\mathrm{e}^{x/2} + \mathrm{e}}{4}$ is $(2, +\infty)$ .

Answer

Solution

Difficulty	7/10
Topics	functions, Extrema, calculus, Substitution, Monotonicity

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