The Second Mathematics Scouts Problem of the Week

Here goes the second Mathematics Scouts Problem of the Week problem.

We again thank every problem proposer.

  • Let a,b be real numbers and a < b. Let f: \mathbb{R} \to \mathbb{R} be a differentiable function with f(a)=0 and \forall x \in (a,b) let 0 \le f'(x) \le 1. Show that \int_{a}^{b} \left( \left ( f(x) \right ) ^3 \right ) dx \le  \left ( \int_{a}^{b} \left ( f(x) \right ) dx \right ) ^2. For the equality is it necessary that the function is constant ? (Proposed by the Mathematics Scouts team). 

Visit here (on AOPS) for more details of this problem. We plan to post the official solution shortly.




For more information see the problem of the week discussion and archive page of the website.




#analysis, #inequalities, #integral-inequalities, #integration, #real-analysis