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Question

Let f:[0,27][13,6] be a differentiable function such that f(x)<0 xDf. If 270xf(x)dx=λ330x2f(x3)dx, then the minimum value of λ is
(Note: Df denotes the domain of the function)

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Solution

270xf(x) dx=λ330x2f(x3) dxλ=270xf(x) dx+330x2f(x3) dx

Putting x3=t3x2dx=dt
λ=270xf(x) dx+270f(t) dtλ=270xf(x)dx+270f(x) dxλ=270[f(x)+xf(x)] dxλ=[xf(x)]270λ=27f(27)

As f(x)<0 xDf, so f(x) is monotonically decreasing function, then
[f(27)]min=13
Therefore, λmin=27×13=9

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