Smallest possible MGF value

HardRandom Variables Statistics~9m

A random variable $X$ has moment generating function $M(\theta) = \mathbb{E}[e^{\theta X}]$ , and $M(1) = 7$ . Over all such $X$ , what is the smallest possible value of $M(4)$ ?

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$\log M(\theta)$ is convex, and $\log M(0)=0$ , $\log M(1)=\log 7$ .
Extrapolate the chord to $\theta=4$ : $\log M(4)\ge 4\log 7$ , i.e. $M(4)\ge 7^4$ (a constant $X$ achieves it).

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Smallest possible MGF value

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