What exactly is the Mean Value Theorem and how does it apply to Rolle's Theorem? (Calculus)
(1) The mean value theorem: Suppose f is a continuous function on an interval [a,b]. Then if `f(a) <= k <= f(b)` for some `k in RR` or `f(a)>=k>=f(b)` , then there must be at least one `c in (a,b)` such that f(c)=k.
In other words, in order to get from f(a) to f(b), the function takes on all values between the two. The function is continuous, so there are no jumps or holes. There must be at least one point c in the interval where f(c) equals the value k.
(2) Rolle's theorem is a special case of the MVT, where f(a) and f(b) have opposite sign, and k=0. In order to get from a positive to a negative, or vice versa, the function must pass through zero.
Notice how important the requirement of continuity is -- if the function is discontinuous on [a,b], the theorems will not apply.