A reverse Jensen inequality for exponentials

Let x_1, ... x_n be real numbers and t_1, ... t_n be nonnegative numbers summing to 1.

A trivial consequence of Jensen's inequality is that

exp(t_1 x_1 + ... + t_n x_n) <= t_1 exp(x_1) + ... + t_n exp(x_n).

I claim that in the other direction, we have the following:

t_1 exp(x_1) + ... + t_n exp(x_n) <= exp( diam(x) + t_1 x_1 + ... + t_n x_n)

where diam(x) = max_i x_i - min_i x_i is the diameter of {x_1, ..., x_n}.

Any proof ideas? (Unlike my previous goof-ups, I can actually prove this one.)