27∫0xf′(x) dx=λ−33∫0x2f(x3) dx⇒λ=27∫0xf′(x) dx+33∫0x2f(x3) dx
Putting x3=t⇒3x2dx=dt
⇒λ=27∫0xf′(x) dx+27∫0f(t) dt⇒λ=27∫0xf′(x)dx+27∫0f(x) dx⇒λ=27∫0[f(x)+xf′(x)] dx⇒λ=[xf(x)]270⇒λ=27f(27)
As f′(x)<0 ∀x∈Df, so f(x) is monotonically decreasing function, then
[f(27)]min=13
Therefore, λmin=27×13=9