$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$. 

Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ?

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<sub>Source: [AoPS][1]</sub>


  [1]: https://artofproblemsolving.com/community/c7h1570936