InterviewSolution
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InterviewSolution
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Let C_(1) and C_(2) be the graphs of the functions y=x^(2) and y=2x, respectively, where 0le x le 1." Let "C_(3) be the graph of a function y=f(x), where 0lexle1, f(0)=0. For a point P on C_(1), let the lines through P, parallel to the axes, meet C_(2) and C_(3) at Q and R, respectively (see figure). If for every position of P(on C_(1)), the areas of the shaded regions OPQ and ORP are equal, determine the function f(x). |
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Answer» <P> Solution :Let P be on `C_(1),y=x^(2) be (t,t^(2))``therefore"y co-ordinate of Q is also "t^(2)` `"Now, Q on y =2x where "y=t^(2)` `therefore""x=t^(2)//2` `therefore""Q((t^(2))/(2),t^(2))` For POINT R, x=t and it is on y=f(x) `therefore""R(t,f(t))` Given that, Area OPQ = Area OPR `rArr""int_(0)^(t^(2))(sqrt(y)-(y)/(2))dy=int_(0)^(t)(x^(2)-f(x))dx` DIFFERENTIATING both sides w.r.t. t, we GET `(sqrt(t^(2))-(t^(2))/(2))(2t)=t^(2)-f(t)` `rArr""f(t)=t^(3)-t^(2)` `rArr""f(x)=x^(3)-x^(2)` |
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