if a function f is continuous on the closed interval [a, b], where a < b, and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that
 f`(c) =f(b)-f(a)/b-a.
						

							Mean Value Theorem. Let f be a function which is differentiable   on the closed interval [a, b]. Then there exists a            
point c in (a, b) such that

Corollary.
Let f be a differentiable function such that the derivative f ` is positive
on the closed interval [a, b]. Then f is increasing on [a, b].

Let f be a differentiable function such that the derivative f ` is negative
on the closed interval [a, b]. Then f is decreasing on [a, b].


Discussion   
[Using Flash]


First Derivative Test. Suppose that c is a critical point of
the function f and suppose that there is an interval (a, b) containing c.

If f `(x) > 0 for all x in (a, c) and f `(x) < 0 for all x in (c, b), then c is a local maximum of f.

If f `(x) < 0 for all x in (a, c) and f `(x) > 0 for all x in (c, b), then c is a local minimum of f.
						

							Mean Value Theorem. Let f be a function which is differentiable   on the closed interval [a, b]. Then there exists a            
point c in (a, b) such that





Corollary.
Let f be a differentiable function such that the derivative f ` is positive
on the closed interval [a, b]. Then f is increasing on [a, b].

Let f be a differentiable function such that the derivative f ` is negative
on the closed interval [a, b]. Then f is decreasing on [a, b].


Discussion   
[Using Flash]


First Derivative Test. Suppose that c is a critical point of
the function f and suppose that there is an interval (a, b) containing c.

If f `(x) > 0 for all x in (a, c) and f `(x) < 0 for all x in (c, b), then c is a local maximum of f.

If f `(x) < 0 for all x in (a, c) and f `(x) > 0 for all x in (c, b), then c is a local minimum of f.