If and g(x) = cases f(t) : -2 t x, -2 x 0 f(t) : 0 t x, 0 x 3 cases

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions

Problem

f(x) = x^2 - 2|x|

and

g(x) = \begin{cases} \min\{f(t)\} : -2 \le t \le x, -2 \le x \le 0 \\ \max\{f(t)\} : 0 \le t \le x, 0 \le x \le 3 \end{cases}

(i) Draw the graph of

f(x)

and discuss its continuity and differentiability. (ii) Find and draw the graph of

g(x)

. Also, discuss the continuity.