Step By Step Calculus » 10.6 - Mean Value Theorem

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The derivative can tell you a lot about the behavior of a function. The points where f^\prime(x)=0f^\prime(x)=0 or f^\prime(x)f^\prime(x) does not exist are called critical points, and idenitfy the only possible points where a piecewise differentiable function can change between increasing and decreasing.
There are some very helpful theorems relating critical points, and the sign of the derivative, to the behaviour of a function . The first is Rolle’s theorem.
  • It states that if ff is continuous on [a,b][a,b], differentiable on (a,b)(a,b), and satisfies f(a)=f(b)f(a)=f(b), then there is a c\in (a,b)c\in (a,b) such that f^\prime(c)=0f^\prime(c)=0.
Rolle’s Theorem implies, among other things, that two functions with the same derivative differ by at most a constant. It also implies the Mean Value Theorem, which is a key result connecting the sign of the derivative to the increasing or decreasing nature of the function.
  • The Mean value theorem states that if ff is continuous on [a,b][a,b], and differentiable on (a,b)(a,b) then there is a c\in (a,b)c\in (a,b) such that f(b)-f(a)=f^\prime(c)(b-a)f(b)-f(a)=f^\prime(c)(b-a).
Suppose ff is continuous on \left[ a,b\right] \left[ a,b\right] . Then it follows from the Mean Value Theorem that
$ \begin{array}{lll} \multicolumn{1}{c}{\text{Condition}} & &\multicolumn{1}{c}{\text{Result}} \\ f^{\prime }(x)=0\text{ on }(a,b) & &f\text{ is constant on }\left( a,b\right) \\ f^{\prime }(x)>0\text{ on }(a,b) & &f\text{ is increasing on }\left( a,b\right) \\ f^{\prime }(x)<0\text{ on }(a,b) & &f\text{ is decreasing on }\left( a,b\right) \end{array} $