If `m` is the least value (global minimum) and `M` is the greatest value (global maximum) of the function `f(x)` on the interval `[a , b]` (estimation of an integral), then `m(b-a)lt=int_a^bf(x)dxlt=M(b-a)dot`

Find the absolute maximum value and the absolute minimum value of the functions in the given intervals: f(x) = x^(3), x in [– 2, 2]

Find the absolute maximum value and the absolute minimum value of the functions in the given intervals: f (x) = sin x + cos x , x in [0,pi]

Find the absolute maximum value and the absolute minimum value of the functions in the given intervals: f(x) = (x-1)^(2) +3, x in [-3 ,1]

If M and m are the maximum and minimum values respectively of the function f(x)= x+ (1)/(x) , then the value of M - m is-

Evaluate the following integrals using the definition of definite integral as the limit of a sum : int_(a)^(b)x^(m)dx[mne-1]

Evaluate: inta^(m x)b^(n x)dx

The least value of m which makes the roots of the equation x^2+5x+m=0 imginary is-

If x in [2,3] , then the value of c in Lagrange's M.V.T. for the function f(x) =x+(1)/(x) is -

STATEMENT 1: If f(x) is continuous on [a , b] , then there exists a point c in (a , b) such that int_a^bf(x)dx=f(c)(b-a) STATEMENT 2: For a < b , if ma n dM are, respectively, the smallest and greatest values of f(x)on[a , b] , then m(b-a)lt=int_a^bf(x)dxlt=(b-a)Mdot

Let f be a continuous function on [a ,b]dot Prove that there exists a number x in [a , b] such that int_a^xf(t)dt=int_x^bf(t)dtdot