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Find the continuous function f where (x^...

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`

Text Solution

Verified by Experts

The correct Answer is:
`f(x)=1/(k+1)[x^4-kx^3+(2k-4)x^2]`
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Find the continuous function f where (x^(4)-4x^(2))<=f(x)<=(2x^(2)-x^(3)) such that the area bounded by y=f(x),y=x^(4)-4x^(2) then y-axis,and the line x=t, where (0<=t<=2) is k xx the area bounded by y=f(x),y=2x^(2)-x^(3),y-axis, and line x=t(where0<=t<=2).

If the area bounded by y=f(x), the x -axis, the y-axis and x=t(t>0) is t^(2), then find f(x) .

Knowledge Check

  • Area bounded by the parabola x^(2)=-4y , the X-axis and the lines x = 0, x = 4 is

    A
    `(16)/(3)`
    B
    `(8)/(3)`
    C
    8
    D
    6

  • The area bounded by the curve y^(2) = 4x and the line 2x-3y+4=0 is

    A
    `1/3`
    B
    `2/3`
    C
    `4/3`
    D
    `5/3`

  • The area bounded by the parahola y^(2)=4x and x+y=3 is

    A
    `16/3`
    B
    `32/3`
    C
    `64/3`
    D
    None of these

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