(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...

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(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.