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))`.

If the function f:[0,8]toR is differentiable, then for 0ltalphalt1 and 0lt beta lt 2, int_(0)^(8)f(t)dt is equal to

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))

Prove that: cos2alpha\ cos2beta+sin^2(alpha-beta)-sin^2(alpha+beta)=cos2(alpha+beta) .

(a) show that alpha^2 + beta^2 + alpha beta =0

If alpha, beta are the rootsof 6x^(2)-4sqrt2 x-3=0 , then alpha^(2)beta+alpha beta^(2) is

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

Let f(x) be a differentiable function and f(alpha)=f(beta)=0(alpha< beta), then the interval (alpha,beta)

If alpha,beta are the roots of the equation x^(2)+px+q=0 , find the value of (a) alpha^(3)beta+alphabeta^(3) (b) alpha^(4)+alpha^(2)beta^(2)+beta^(4) .

Let f(x) is a quadratic expression with positive integral coefficients such that for every alpha, beta in R, beta lt alpha,int_alpha^beta f(x) dx lt 0 . Let g(t) = f''(t) f(t), and g(0) = 12 , then