Problem

The diagram shows the points $A\big(1\frac12, 5\frac12\big)$ and $B\big(7\frac12, 3\frac12\big)$ lying on the curve with equation $y = 9x - (2x + 1)^{\frac32}$.

(a) Find the coordinates of the maximum point of the curve.

(b) Verify that the line $AB$ is the normal to the curve at $A$.

(c) Find the area of the shaded region.