Problem

The diagram shows the curve with equation

$$y = 2x^{-\frac{2}{3}} - 3x^{-\frac{1}{3}} + 1$$

for $x > 0$. The curve crosses the $x$-axis at points $A$ and $B$ and has a minimum point $M$.

(a) Find the exact coordinates of $M$.

(b) Find the area of the region bounded by the curve and the line segment $AB$.