Let `C_1C_2`
be the graphs of the functions `y=x^2` and `y=2x ,`
respectively, where `0lt=xlt=1.`
Let `C_3`
be the graph of a function `y=f(x),`
where `0lt=xlt=1, f(0) =0.`
For a point `P` on `C_1,`
let the lines through `P ,`
parallel to the axis, meet `C_2` and `C_3`
at `Q` and `R ,`
respectively (see Figure). If for every position of `P(onC_1),`
the areas of the shaded regions `OPQ` and `ORP`
are equal, determine the function `f(x)`
Text Solution
Verified by Experts
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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