Mean Value Theorem
If f is continuous on [a,b] and differentials on (a,b), then there is a number c such thatf'(c)=(f(b)-f(a))/(b-a)
The Mean Value Theorem states that for a differentiable function f(x) on an interval [a,b], there exists a point c in (a,b) such that:
f'(c) = (f(b) – f(a))/(b – a)
In other words, the slope of the tangent line at point c is equal to the average rate of change of the function over the interval [a,b].
This theorem is important because it allows us to make conclusions about the behavior of the function over the interval based on the behavior at a single point. For example, if we know that the average rate of change of a function over an interval is zero, then we can conclude that there must be at least one point in the interval where the function has a horizontal tangent line (i.e. f'(c) = 0).
The Mean Value Theorem also forms the basis for many other results in calculus, such as the derivative test for finding extrema and the Fundamental Theorem of Calculus.
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