The mean value theorem connects the average rate of change of a function to its derivative. It says that for any differentiable function ff and an interval [a,b]open bracket, a, comma, b, close bracket (within the domain of ff), there exists a number cc within (a,b)left parenthesis, a, comma, b, right parenthesis such that f′(c)f, prime, left parenthesis, c, right parenthesis is equal to the function's average rate of change over [a,b]open bracket, a, comma, b, close bracket.
Graphically, the theorem says that for any arc between two endpoints, there's a point at which the tangent to the arc is parallel to the secant through its endpoints.
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Check your understanding
f(x)=x3−6x2+12xf, left parenthesis, x, right parenthesis, equals, x, start superscript, 3, end superscript, minus, 6, x, start superscript, 2, end superscript, plus, 12, x
Let cc be the number that satisfies the Mean Value Theorem for ff on the interval [0,3]open bracket, 0, comma, 3, close bracket.