1. Do Now: Find all extrema of 𝑓 𝑥 = 1 3 𝑥 3 + 𝑥 2 −3𝑥+2 on [-4,3]

2. 3.1 Part II
The Mean Value Theorem

3. Mean value theorem
If y = f(x) is continuous at every point on 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:
In other words, there is at least one point where the instantaneous rate of change equals the average rate of change.

4. Using the MVT 1) Find 2 ordered pairs using the x-values given.
2) Find the slope between those points (avg. rate of change)
3) Find 𝑓 ′ 𝑥 and set it equal to the number you found in (2).
4) Solve for x.

5. Example 1 Avg rate of change = 𝑓 2 −𝑓(0) 2−0 = 4−0 2 =2
Show that the function 𝑓 𝑥 = 𝑥 2 satisfies the hypothesis of the Mean Value Theorem on the interval [0,2]. Then find the value of c (where the average rate of change = the instantaneous rate of change.
𝑓(𝑥) is continuous on [0,2] and differentiable on (0,2)
Avg rate of change = 𝑓 2 −𝑓(0) 2−0 = 4−0 2 =2
Instantaneous rate of change
𝑓 ′ 𝑥 =2𝑥
Set 2𝑐=2
𝑐=1

6. example 2
Explain why each of the following functions fail to satisfy the conditions of the MVT on the interval [–1, 1].
𝑓 𝑥 = 𝑥 +1

7. Rolle’s Theorem Rolle’s Theorem is a special case of the MVT:
If y = f(x) is continuous at every point on the closed interval [a, b] and differentiable at every point of its interior (a, b), and if f(a) = f(b) = 0, then there is at least one point c in (a, b) such that f ′(c) = 0.

8. Example 4
Type equation here. 
𝑓 ′ 𝑥 =−2𝑥=0
𝑥=0