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.

1.\(f(x) = 5 - 12x + 3{x^2}\), \(\left( {1,3} \right)\)

Short answer

There exist a required number\(c = 2\), which belongs to the open interval\((1,3)\)such that\({f^\prime }(2) = 0\).

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

Step-by-step solution

Given function

The function \(f(x) = 5 - 12x + 3{x^2}\) 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\(f(x) = 5 - 12x + 3{x^2}\)is polynomial function, it is continuous on\(R\).

Therefore, the function\(f(x)\)is continuous on the closed interval\(\left( {1,3} \right)\).

Note that, the function\(f(x) = 5 - 12x + 3{x^2}\)is a polynomial function, it is differentiable everywhere.

Thus, the function \(f(x)\)is differentiable on the open interval \(\left( {1,3} \right)\).

Check the third condition of Rolle’s Theorem

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

From the given interval, it is observed that\(a = 1\)and\(b = 3\).

Substitute\(1\)for\(x\)in\(f(x)\).

\(\begin{aligned}{c}f(1) &= 5 - 12(1) + 3{(1)^2}\\ &= 5 - 12 + 3\\ &= - 4\end{aligned}\)

Substitute,\({\rm{x }} \to 3\)in\(f(x)\).

\(\begin{aligned}{c}f(3) &= 5 - 12(3) + 3{(3)^2}\\ &= 5 - 36 + 27\\ &= - 4\end{aligned}\)

Hence,\(f(1) = f(3)\), 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\((1,3)\).

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

\(\begin{aligned}{c}{f^\prime }(x) &= \frac{d}{{dx}}\left( {5 - 12x + 3{x^2}} \right)\\ &= - 12 + 6x\end{aligned}\)

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

\(\begin{aligned}{c} - 12 + 6c &= 0\\6c &= 12\\c &= 2\end{aligned}\)

Notice that there exist a required number\(c = 2\), which belongs to the open interval\((1,3)\)such that\({f^\prime }(2) = 0\).

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