The

**Mean Value Theorem** establishes a relationship between the slope of a tangent line to a curve and the secant line through points on a curve at the endpoints of an interval. The theorem is stated as follows.

If a function *f(x)* is continuous on a closed interval [a,b] and differentiable on an open interval (a,b), then at least one number c ∈ (a,b) exists such that

** Figure 1 **The Mean Value Theorem.

Geometrically, this means that the slope of the tangent line will be equal to the slope of the secant line through *(a,f(a))* and *(b,f(b))* for at least one point on the curve between the two endpoints. Note that for the special case where *f(a) = f(b)*, the theorem guarantees at least one critical point, where *f(c)* = 0 on the open interval ( *a, b*).

**Example 1**: Verify the conclusion of the Mean Value Theorem for *f(x)*= *x* ^{2}−3 *x*−2 on [−2,3].

The function is continuous on [−2,3] and differentiable on (−2,3). The slope of the secant line through the endpoint values is

The slope of the tangent line is

Because ½ ∈ [−2,3], the *c* value referred to in the conclusion of the Mean Value Theorem is *c* = ½