Bit of trouble differentiating this one. Can anyone help? Many thanks.
Q. A curve has equation y = x2. 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 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 = x2, then p = (x, x2)
(ii) If bp ab, we can infer that b = (x,0).
Area of a triangle apb on a plane is: [(y2 - y1)(x1 - x3) - (x2 - x1)(y1 - y3)] => [(x2 - 0)(6 - x) - (x - 6)(0 - 0)] => [6x2 - x3]
= (12x - 3x2) = 0 => 12x - 3x2 = 0 => 4x - x2 = 0 => x2 = 4x => x = 4
Ans: (From text book): x = 2