Let's do some calculus! (53)

Let $f: \mathbb{R} \mapsto \mathbb{R}$ be a function continuous in $[1,6]$ and differentiable in $(1,6)$ such that $f(1) = 3$ and $f(6) = 5$ . It is given that $\displaystyle \int_1^6 (f(x))^2 \,dx = 11$ , then what is the minimum value of

$\large \left( \int_1^6 x^2 (f(x))^2 \,dx \right) \left( \int_1^6 (f'(x))^2 \,dx \right)$

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