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Found problems: 2

2015 Romania National Olympiad, 3
Tags: integral calculus , integral inequality , calculus
Let $\mathcal{C}$ be the set of all twice differentiable functions $f:[0,1] \to \mathbb{R}$ with at least two (not necessarily distinct) zeros and $|f''(x)| \le 1,$ for all $x \in [0,1].$ Find the greatest value of the integral $$\int\limits_0^1 |f(x)| \mathrm{d}x$$ when $f$ runs through the set $\mathcal{C},$ as well as the functions that achieve this maximum. [i]Note: A differentiable function $f$ has two zeros in the same point $a$ if $f(a)=f'(a)=0.$[/i]

2024 CMI B.Sc. Entrance Exam, 1
Tags: geometry , integral calculus , calculus
(a) Sketch qualitativly the region with maximum area such that it lies in the first quadrant and is bound by $y=x^2-x^3$ and $y=kx$ where $k$ is a constent. The region must not have any other lines closing it. Note: $kx$ lies above $x^2-x^3$ (b) Find an expression for the volume of the solid obtained by spinning this region about the $y$ axis.