Bit of trouble differentiating this one. Can anyone help? Many thanks.A curve has equation y = x

Q.^{2}. Three points form the triangle apb. p(x,y) is a point on the curve, a, on the x-axis, is (6,0) & a final point b, also on the x-axis. bp $\displaystyle \perp$ ab. (i) Express the coordinates of p, in terms of x only, (ii) Find the value of x if the area of the triangle abp is a maximum and hence find this maximum area.

Attempt:(i) If y = x^{2}, then p = (x, x^{2})

(ii) If bp $\displaystyle \perp$ ab, we can infer that b = (x,0).

Area of a triangle apb on a plane is: $\displaystyle \frac{1}{2}$[(y_{2}- y_{1})(x_{1 }- x_{3}) - (x_{2}- x_{1})(y_{1}- y_{3})] => $\displaystyle \frac{1}{2}$[(x^{2}- 0)(6 - x) - (x - 6)(0 - 0)] => $\displaystyle \frac{1}{2}$[6x^{2}- x^{3}]

$\displaystyle \frac{dA}{dx}$ = $\displaystyle \frac{1}{2}$(12x - 3x^{2}) = 0 => 12x - 3x^{2}= 0 => 4x - x^{2}= 0 => x^{2}= 4x => x = 4

Ans:(From text book): x = 2