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