Given a function `f:[0,4]toR` is differentiable ,then prove that for some `alpha,beta in (0,2), int_(0)^(4)f(t)dt=2alphaf(alpha^(2))+2betaf(beta^(2))`.

Explanation

`I=int_(0)^(4)f(t)dt,` put `t=x^(2)`
`:.I=2int_(0)^(2)xf(x^(2))dx`
using LMVT, we have
`(int_(0)^(2)2xf(x^(2))dx-int_(0)^(0)2xf(x^(2))dx)/(2-0)=2yf(y^(2))` for somd `yepsilon(0,2)`
`impliesint_(0)^(2)2xf(x^(2))dx=2.2yf(y^(2))=2{(2 alpha f(alpha^(2))+2betaf(beta^(2)))/2}`
[where `0lt beta lt y lt alpha lt 2`. and using intermediate Vlue Theorem]
`implies int_(0)^(2)2xf(x^(2))dx=2alpha f(alpha^(2))+2betaf(beta^(2))`