Find a continuous function f, where `(x^(4)-4x^(2))lef (x) le (2x^(2)-x^(3))` such that the area bounded by `y=f(x), y=x^(4)-4x^(2),` the y-axis, and the line `x=t," where "(0letle2)` is k times the area bounded by `y=f(x),y=2x^(2)-x^(3),` y-axis, and line `x=t("where "0le t le 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)).`