4.6 The Mean Value Theorem
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Rolle’s Theorem
Let f be a continuous on the closed interval[a,b] and differentiable on the open
interval (a,b) If f(a) = f(b) then there is at least one number c in (a,b) such that
f (c)  0 .
The Mean Value Theorem
If f is continuous on the closed interval [a,b] and differentiable on the open interval
(a,b), then there exists a number c such that
f (c) 
f (b)  f (a )
ba
Example 1 Given 𝑝(𝑥) = 𝑥 3 − 7𝑥 2 + 10𝑥, find an interval on which Rolle’s
Theorem applies. Then find all values of c where 𝑓 ′ (𝑐) = 0.
Example 2 Given 𝑝(𝑥) = 𝑥 3 − 7𝑥 2 + 10𝑥 on [0,1], show that the Mean Value
Theorem applies.
Example 3
A plane begins its takeoff at 2:00 pm on a 2500-mile flight. The plane arrives at its
destination at 7:30 pm. Explain why there were at least two times during the flight
when the speed of the plane was 400 miles per hour.
Example 4
When an object is removed from a furnace it has a temperature of 1500°F and is
placed in a 90°F room to cool. After five hours, the temperature of the object is
390°F. Explain why there must be at least one time in the interval where the
temperature is decreasing at a rate of 222°F per hour.
HW p306 1,7,11,13,15,17,27,33,41