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))`.

f: [0.4]-> R is a differentiable function. Then prove that for some a,bin(0,4) , f^2(4)-f^2(0)=8f'(a)*f(b) .

Let f(x) be a differentiable function in the interval (0, 2) then the value of int_(0)^(2)f(x)dx

Let f is a differentiable function such that f'(x) = f(x) + int_(0)^(2) f(x) dx, f(0) = (4-e^(2))/(3) , find f(x).

Let f:RtoR be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt . f(x) increases for

If sin(theta+alpha)=aandsin(theta+beta)=b , then prove that cos2(alpha-beta)-4abcos(alpha-beta)=1-2a^(2)-2b^(2) .

Let y = f(x), f : R ->R be an odd differentiable function such that f'''(x)>0 and g(alpha,beta)=sin^8alpha+cos^8beta+2-4sin^2alpha cos^2 beta If f''(g(alpha, beta))=0 then sin^2alpha+sin^2beta is equal to

If cosalpha+cosbeta=0=sinalpha+sinbeta, then prove that cos2alpha+cos2beta=-2cos(alpha+beta) .

If alpha and beta are roots of the equation acostheta+bsintheta=c then prove that, sin(alpha+beta)=(2ab)/(a^(2)+b^(2))

If alpha,beta be the roots of the equation 3x^2+2x+1=0, then find value of ((1-alpha)/(1+alpha))^3+((1-beta)/(1+beta))^3

If alpha and beta are the roots of the equation x^2-4x + 1=0(alpha > beta) then find the value of f(alpha,beta)=(beta^3)/2csc^2(1/2tan^(- 1)(beta/alpha))+(alpha^3)/2sec^2(1/2tan^- 1(alpha/beta))