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)).

If the function f : [0,8] to R is differentiable, then for 0 < alpha <1 < beta < 2 , int_0^8 f(t) dt is equal to (a) 3[alpha^3f(alpha^2)+beta^2f(beta^2)] (b) 3[alpha^3f(alpha)+beta^2f(beta)] (c) 3[alpha^3f(alpha^2)+beta^2f(beta^3)] (d) 3[alpha^2f(alpha^3)+beta^2f(beta^3)]

Prove that (cos alpha+cos beta)^(2)+(sin alpha -sin beta)^(2)=4 cos^(2) ((alpha+beta)/(2))

If alpha and beta are the roots of x^(2)+6x-4=0 , find the value of (alpha-beta)^(2) .

If int_(0)^(x) f(t)dt=x^2+int_(x)^(1) t^2f(t)dt , then f'(1/2) is

If int_(0)^(x)f(t)dt=e^(x)-ae^(2x)int_(0)^(1)f(t)e^(-t)dt , then

If alpha and beta are the roots of the equation 3x^(2) - 5x + 2 = 0 , find the value of (i) (alpha)/(beta) + (beta)/(alpha) (ii) alpha-beta (iii) (alpha^(2))/(beta) + (beta^(2))/(alpha)

If alpha and beta are the roots of x^(2)+6x-4=0 , find the values of (alpha-beta)^(2) .

If alpha and beta are the roots of 2x^(2) – 3x -4 = 0 find the value of 2 alpha^(2)+ beta^(2)

alpha, beta are roots of y^(2)-2y-7=0 find, (i) alpha^(2)+beta^(2) " (ii) " alpha^(3)+beta^(3)