How can you maximize your happiness under a budget? When does a function reach its minimum value? When does a curve change direction? The calculus of extrema explains these "extreme" situations.

*Source: Mandelbrot #2*

If \(S = \dfrac{a\sqrt{b}}{c},\) 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

\[\large {x}^{2}-{2}^{x}=0? \]

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

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