Concavity of a function related to the Brunn-Minkowski inequality

Question

I'm reading something on Brunn-Minkowski inequality and I've come along the following fact: the inequality, which says that if K and L are convex bodies then $$V((1-\lambda)K+\lambda L)^{1/n}\geq (1-\lambda)V(K)^{1/n}+\lambda V(L)^{1/n}$$ implies that the function $$f_1(t)=V((1-t)K+tL)^{1/n}$$ is concave in [0,1]. Why? I can see the reason why $f_2(K)=V(K)^{1/n}$ is convex in the space of convex bodies equipped with Minkowski sum, but the concavity of $f_1:[0,1]\rightarrow \mathbb{R}$ is not clear to me.