Given $f: \mathbb{R^3} \rightarrow \mathbb{R}\in C^2$ s.t. $∆f > 0$ prove that $\frac{d\int_{\partial B(0,r)}f}{dr} > 0$

Question

Given $f: \mathbb{R^3} \rightarrow \mathbb{R}\in C^2$ s.t. $∆f > 0$ prove that $\frac{d\int_{\partial B(0,r)}f}{dr} > 0$.

I know that this is just using Green's first formula, but I'm having a little bit of trouble applying it. I know that $\int_{\partial B(0,r)}Df_N > 0$, but I can't seem to formally prove that that is enough for this question. Any help would be appreciated.