Section 4.2
Mean Value Theorem
Mean Value Theorem
 If y = f(x) is continuous at every point of the closed
interval [a,b] and differentiable at every point of its
interior (a,b), then there is at least one point c in
(a,b) at which
′
𝑓 𝑐 =
𝑓 𝑏 −𝑓(𝑎)
.
𝑏 −𝑎
 In other words, if the hypothesis is true, then there
is some point at which the instantaneous rate of
change is equal to the average rate of change for
the interval.
Find the point at which the function
satisfies the Mean Value Theorem
1. f(x) = x3 – 3x + 5
-1 < x < 1
(Be sure to verify that the
conditions are met for using the
MVT)
2. f(x) = xex on the interval [0, 1]
Implications of the MVT
If f’ > 0, then f is…..
Increasing.
If f’ < 0, then f is…..
Decreasing
If f’ = 0, then f is…..
Constant.
Example
Determine where the following
function is increasing, decreasing,
constant.
f(x) = 2x3 – 9x2 + 12x - 5
Another Application of MVT
If f’(x) and g’(x) are equal at each point of
an interval, then f(x) and g(x) are separated
only by a constant.
Example: Find a function whose derivative
is 6x2 + 5