Find the continuous function `f` where `(x^4-4x^2)lt=f(x)lt=(2x^2-x^3)` such that the area bounded by `y=f(x),y=x^4-4x^2dot` then y-axis, and the line `x=t ,` where `(0lt=tlt=2)` is `k` times the area bounded by `y=f(x),y=2x^2-x^3`,y-axis , and line `x=t(w h e r e0lt=tlt=2)dot`

Answer

`(1)/(k+1)(x^(4)-kx^(3)+2(k-2)x^(2))`

Explanation

According to the given conditions
`overset(t)underset(0)int[f(x)-(x^(4)-4x^(2))]dx=koverset(t)underset(0)int[(2x^(2)-x^(3))-f(x)]dx`
Differentiating both sides w.r.t. we get
`f(t)-(t^(4)-4t^(2))=k(2t^(2)-t^(3)-f(t))`
`"or "(1+k)f(t)=k2t^(2)-kt^(2)+t^(4)-4t^(2)`
`rArr" "f(t)=(1)/(k+1)[t^(4)-kt^(3)+(2k-4)t^(2)]`
Hence, required f is given by `f(x)=(1)/(k+1)(x^(4)-kx^(3)+2(k-2)x^(2)).`