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Let \( g \) be a non constant differentiable real function on a finite interval \( [a, b] \) with \( g(a)=g(b)=0 \). Show that there exists a \( c \in (a, b) \) such that:

$$|g'(c)|> \frac{4}{(b-a)^2} \int_a^b |g(t)|\, {\rm d}t$$
(Qualifying Exams, Wisconsin-Madison, 2015)