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

Let f : R rarr R be a differentiable function satisfying f(x) = f(y) f(x - y), AA x, y in R and f'(0) = int_(0)^(4) {2x}dx , where {.} denotes the fractional part function and f'(-3) - alpha e^(beta) . Then, |alpha + beta| is equal to.......

If for some differentiable function f(alpha)=0 and f'(alpha)=0, Statement 1: Then sign of f(x) does not change in the neighbourhood of x=alpha Statement 2:alpha is repeated root of f(x)=0

Let f:[0,4]rarr R be a differentiable function then for some alpha,beta in (0,2)int_(0)^(4)f(t)dt

Let y=f(x) be a thrice differentiable function in (-5,5) .Let the tangents to the curve y=f(x) at (1,f(1)) and (3,f(3)) make angles pi/6 and pi/4 ,respectively with positive "x" -axis.If 27int_(1)^(3)((f'(t))^(2)+1)f''(t)dt=alpha+beta sqrt(3) where alpha,beta are integers,then the value of alpha+beta equals

Let f(x) be an identity function and alpha, beta be the roots of equation x^(2)-5x+9=0 then the value of f(alpha)+f(beta) is equal to

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

Let f(x)=alpha (x) beta (x) gamma (x) for all real x, where alpha (x), beta (X) and gamma (x) are differentiable functions of x. If f'(2)=18 f(2), alpha'(2)=3 alpha (2), beta' (2)=-4 beta (2) and gamma'(2)=k gamma (2) , then the value of k is

Let f(x) be a second degree polynomial function such that f(-1)=f(1) and alpha,beta,gamma are in A.P. Then, f'(alpha),f'(beta),f'(gamma) are in