Calculus
# Extrema

$y=x^2$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?

A circle rests in the interior of the parabola with equation*Source: Mandelbrot #2*

$S$ be the maximum possible area of a right triangle that can be drawn in a semi-circle of radius $1$, where one of the legs (and not the hypotenuse) of the triangle must lie on the diameter of the semicircle.

LetIf $S = \dfrac{a\sqrt{b}}{c},$ where $a,c$ are positive coprime integers and $b$ is a positive square-free integer, find $a + b + c.$

How many real values of $x$ satisfy the equation

$\large {x}^{2}-{2}^{x}=0?$

$least$ $perimeter$ of an **isosceles** triangle in which a circle of radius $\sqrt{3}$ can be inscribed ?

**Note :**
The picture shown is a rough one.