A calculus problem by Priyanshu Mishra

Consider the function $g(y) = 2 \sin y$ . If we wish to maximize the magnitude of $2 \sin y - x$ when $x$ is positive then we want to find the minimum of $2 \sin y$ and then subtract $x$ from it. The minimum of $g(y)$ is just $-2$ so $f(x)$ for $x>0$ is $|-2 - x| = |2 + x| = 2 + x$ . For negative $x$ we want to find the maximum of $2 \sin y$ and then subtract (add magnitude) $x$ from it. The maximum of $g(y)$ is $2$ so we have $|2 - x| = 2-x$ . So a new definition for the function is: $f(x) = \begin{cases} x> 0 & 2 + x\\ x \leq 0 & 2 - x \end{cases}$ and upon closer inspection this function is equal to $f(x) = 2 + |x|$ . Thus the minimum value of $f$ occurs at $x=0$ giving $f(0) = 2$ .