Problem
Can an arc of a parabola inside a circle of radius 1 have a length greater than 4?
Solution
No. Without loss of generality let $y=ax^2$ be the parabola. After a bit, one can be convinced that in order to maximize the arc length inside the circle, it has to have its center at $(0,1)$.
By finding the arc length as a function of $a$ and applying the fundamental theorem of calculus, one can see that the arc length of the parabola inside the circle is an increasing function of $a$. Since for $a=0$ the arc length is 0 and for $a=\infty$ the arc length is 4, there is no such solution for the problem.
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