CIIM 2020 Problem 1

Let $\alpha>1$ and consider the function $f(x)=x^{\alpha}$ for $x \ge 0$ . For $t>0$ , define $M(t)$ as the largest area that a triangle with vertices $(0, 0), (s, f(s)), (t, f(t))$ could reach, for $s \in (0,t)$ . Let $A(t)$ be the area of the region bounded by the segment with endpoints $(0, 0)$ , $(t, f(t))$ and the graph of $y =f(x)$ . (a) Show that $A(t)/M(t)$ does not depend on $t$ . We denote this value by $c(\alpha)$ . Find $c(\alpha)$ . (b) Determine the range of values of $c(\alpha)$ when $\alpha$ varies in the interval $(1, +\infty)$ .