Question

1–4 ■ 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.

4.\(f(x) = \cos 2x\),\(\left( {\frac{\pi }{8},\frac{{7\pi }}{8}} \right)\)

Short answer

The number \(c = \frac{\pi }{2}\) satisfies the conclusion of Rolle's Theorem.

Step-by-step solution

Given function

The function \(f(x) = \cos 2x\) is a polynomial.

Concept of Rolle’s Theorem

Rolle's Theorem:

“Let\(f\)be a function that satisfies the following hypothesis:

1)\(f\)is continuous on the closed interval\(\left( {a,b} \right)\).

2)\(f\)is differentiable on the open interval\((a,b)\).

3)\(f(a) = f(b)\).

Then, there is a number\(c\)in\((a,b)\)such that,\(f'(c) = 0\).”

Check the first and second condition of Rolle’s Theorem

Since the function\(f(x) = \cos 2x\)is cosine function, it is continuous on\(R\).

Therefore, the function\(f(x)\)is continuous on the closed interval\(\left( {\frac{\pi }{8},\frac{{7\pi }}{8}} \right)\).

Note that, the function\(f(x) = \cos 2x\)is a cosine function, it is differentiable everywhere.

Thus, the function \(f(x)\) is differentiable on the open interval \(\left( {\frac{\pi }{8},\frac{{7\pi }}{8}} \right)\).

Check the third condition of Rolle’s Theorem

To satisfy the third condition of the theorem,\(f(a)\)must be equal to\(f(b)\).

From the given interval, it is observed that\(a = \frac{\pi }{8};\;b = \frac{{7\pi }}{8}\).

Substitute, \(\frac{\pi }{8} \to x\)in\(f(x)\).

\(\begin{aligned}{c}f\left( {\frac{\pi }{8}} \right) = \cos 2\left( {\frac{\pi }{8}} \right)\\ = \cos \frac{\pi }{4}\\ = \frac{1}{{\sqrt 2 }}\end{aligned}\)

Substitute,\(\frac{{7\pi }}{8}\)for\(x\)in\(f(x)\).

\(\begin{aligned}{c}f\left( {\frac{{7\pi }}{8}} \right) = \cos 2\\ = \cos \frac{{7\pi }}{4}\\ = \frac{1}{{\sqrt 2 }}\end{aligned}\)

Hence,\(f\left( {\frac{\pi }{8}} \right) = f\left( {\frac{{7\pi }}{8}} \right)\), that is at both points\(f(x)\)attains same functional values.

Therefore,\(f(a) = f(b)\).

Thus, the function \(f(x)\)satisfies the conditions of Rolle's Theorem.

Finding the values of number c

Obtain the number that satisfies the conclusion of Rolle's Theorem.

Since the number c satisfies the conditions of Rolle's Theorem, it should lie on the open interval\(\left( {\frac{\pi }{8},\frac{{7\pi }}{8}} \right)\).

Obtain the derivative of\(f(x)\).

\(\begin{aligned}{c}{f^\prime }(x) = \frac{d}{{dx}}(\cos 2x)\\ = - 2\sin 2x\end{aligned}\)

Replace\(x\)by\({\mathop{\rm c}\nolimits} \)in\({f^\prime }(x)\)and set\({f^\prime }(c) = 0\)to obtain the value of\({\mathop{\rm c}\nolimits} \).

\(\begin{aligned}{c} - 2\sin 2c = 0\\c = \frac{{n\pi }}{2}\end{aligned}\)

Notice that there exist a required number that is\(\frac{\pi }{2}\), which belongs to the open interval\(\left( {\frac{\pi }{8},\frac{{7\pi }}{8}} \right)\)such that\({f^\prime }\left( {\frac{\pi }{2}} \right) = 0\).

Therefore, it can be concluded that the number \(c = \frac{\pi }{2}\) satisfies the conclusion of Rolle's Theorem.