Verify that the function satisfies the three hypotheses of Rolle's Theorem

Question

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.

f(x) = cos 2x, [π/8, 7π/8]

Answer 1

The three conditions for Rolle's Theorem to hold on an interval [a,b] are

 

(i) f(x) is continuous on [a,b]

(ii) f(x) is differentiable on (a,b)

(iii) f(a) = f(b)

 

So, let's prove that the function satisfies these conditions.

(i) cos (2x) is continuous on (-∞, ∞), so it is also continuous on [pi/8, 7*pi/8].

(ii) The derivative of f(x) is -2sin(2x), which exists on [pi/8, 7*pi/8].

(iii) f(pi/8) = cos (pi/4) = √(2)/2

f(7*pi/8) = cos (7*pi/4) = √(2)/2.

 

So, the function satisfies the 3 hypothesis.

Next, we are guaranteed that there is at least one number c∈(pi/8, 7*pi/8) such that f'(c) = 0.  So, let's solve:

 

f'(x) = -2sin(2x)

-2sin(2x) = 0

sin(2x) = 0

 

A variable substitution may be helpful:

Let u = 2x. We are given that pi/8 < x < 7*pi/8, which means that

pi/4 < u < 7*pi/4. 

 

On this interval, sin(u) = 0 has the single solution u = pi, which means

2x = pi

x = pi/2.

 

So, there is one c in (pi/8, 7*pi/8) which satisfies the theorem, which is c = pi/2.