Your question says nothing about the triangle area...
Bit of trouble differentiating this one. Can anyone help? Many thanks.
Q. A curve has equation y = x^{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 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 ab, we can infer that b = (x,0).
Area of a triangle apb on a plane is: [(y_{2} - y_{1})(x_{1 }- x_{3}) - (x_{2} - x_{1})(y_{1} - y_{3})] => [(x^{2} - 0)(6 - x) - (x - 6)(0 - 0)] => [6x^{2} - x^{3}]
= (12x - 3x^{2}) = 0 => 12x - 3x^{2} = 0 => 4x - x^{2} = 0 => x^{2} = 4x => x = 4
Ans: (From text book): x = 2
If B is allowed to be located right of A, then there is no maximum triangle area. If B has to be left of A, then I agree that the maximum is reached when x = 4. When x = 2, the area is 8, but when x = 4, the area is 16.