$(\prod_1^n (1+c_i t))^\frac{1}{n}$ is concave in t

Question

I would like to show that $f(t) = (\prod_{i=1}^n (1+c_i t))^\frac{1}{n}$ is concave on $[0,1]$.

I think it is related to AM-GM inequality. $f(t)$ is GM, while corresponding AM is linear and is always above $f(t)$.

Also it is easy to show that $f(\alpha t)\geq \alpha f(t)$.

I am not sure what to do next.