Putnam 2019 A6

Let $g$ be a real-valued function that is continuous on the closed interval $[0,1]$ and twice differentiable on the open interval $(0,1)$. Suppose that for some real number $r>1$, $$\lim_{x \to 0^+} \frac{g(x)}{x^r} = 0.$$ Prove that either $$\lim_{x \to 0^+} g'(x) = 0 \text{or} \limsup_{x \to 0^+} x^r |g''(x)| = \infty.$$