Exponential Integral?

If we define $g(x) \; = \; 2x^2 f(x) - e^x \hspace{2cm} x > 0$ then $g(2) = 0$ and $g'(x) \; = \; 2x^2 f'(x) + 4x f(x) - e^x \; = \; 2x^{-1}e^x - e^x \; = \; \frac{2-x}{x}e^x$ and hence $g(x)$ is increasing for $0 < x< 2$ and decreasing for $x > 2$ . Thus it follows that $g(x) \le 0$ for all $x > 0$ . Since $x^3 f'(x) + g(x) \; = \; 0$ for all $x > 0$ , we deduce that $f'(x) \ge 0$ for all $x > 0$ . Thus $f$ is an increasing function, and therefore has no maxima or minima. The turning point at $x=2$ is a point of inflection.