The Mean Value Theorem for integrals tells us that, for a continuous function f(x), there’s at least one point c inside the interval [a,b] at which the value of the function will be equal to the average value of the function over that interval. This means we can equate the average value of the function over the interval to the value of the function at the single point.
Read MoreThe Mean Value Theorem tells us that, as long as the function is continuous (unbroken) and differentiable (smooth) everywhere inside the interval we’ve chosen, then there must be a line tangent to the curve somewhere in the interval, which is parallel to this line we’ve just drawn that connects the endpoints.
Read MoreRolle’s Theorem can prove all of the following: 1) The existence of a horizontal tangent line in the interval, 2) A point at which the derivative is 0 in the interval, 3) The existence of a critical point in the interval, and 4) A point at which the function changes direction in the interval, either from increasing to decreasing, or from decreasing to increasing.
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