rolles theorem and mean value theorem

Rolle's Theorem - Let f be a real valued function in [a,b] such that f is continuous in [a,b]. f is differentiable in (a,b..

Let f be a real valued function in [a,b] such that f is continuous in [a,b]. f is differentiable in (a,b..

Let f be real valued function in [a,b] such that, 1. f is continuous in [a,b]. 2. f is differentiable in (a,b). ..

Which of the following functions satisfies the mean value theorem? => f ( x ) = x 2 - 6 , x ∈ [7, 8] or f ( x ) = log x , x ∈ [-7, 8] or f ( x ) = [ x ], x ∈ [-7, 7] or f ( x ) = | x |, x ∈ [-8, 8]..

Let f be continuous on [a, b] and differentiable on the open interval (a, b). Then (a) f is increasing on [a, b] if f '(x) > 0 for each x (a, b) (b) f is decreasing on [a, b] if f '(x) < 0 for each x (a, b) This theorem can be proved by using Mean Value ..

Let f be a continuous function on an interval I = [a, b]. Then, f has the absolute maximum value and f attains it at least once in I. Also, f has the absolute minimum value and attains it at least once in ..

Let f be a differentiable function on I and let x 0 be any interior point of I. Then (a) If f attains its absolute maximum value at x 0 , then f ' (x 0 )= 0 (b) If f attains its absolute minimum value at x 0 , then f '(x 0 ) = 0. In view of the above theorems, we sta..

If f(x) is a polynomial in x and is divided by x-a; the remainder is the value of f(x) at x = a i.e., Remainder = f(a..

Let f be real valued function in [a,b] such that, f is continuous in [a,b]. f is differentiable in (a,b)...

Gauss' Theorem - We have already learnt to find the electric field intensity due to a charged conductor using Coulomb's law. Gauss' theorem can also be used to calculate the electric field intensity provided there is a symmetry in the charge distribution. Can be considered as an..