1.

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-a xi s ,`and line `x=t(w h e r e0lt=tlt=2)dot`

Answer» Correct Answer - `(1)/(k+1)(x^(4)-kx^(3)+2(k-2)x^(2))`
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)).`


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