Statement: If f(x) is

(i) continuous in the closed interval [a, b];

(ii) differentiable in the open interval (a, b).

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

Then, there exists at least one number \(c\in (a, b)\) such that f '(c) = 0.

#### Geometrical Meaning of Roll’s Theorem

Let f(x) be a continuous function in an interval [a,b] with A=f(a) and B=f(b).

Then Role’s Theorem asserts that there exist at least one point c lying between A and B such that the tangent at that point is parallel to x-axis.

There may exist more than one points \(c_1, c_2\) and may be more between A and B, so that the tangents at those points are parallel to x-axis

#### The condition where Rolle’s theorem fails

The condition where Rolle’s theorem is NOT applicable, of the Rolle’s theorem fails.Discontinuous | Discontinuous | Not Differentiable |

#### Rolle’s theorem fails: Counter Examples

#### Rolle’s theorem :History

The Roll's theorem says that, if a continuous curve passes through the same y-value (such as the x-axis) twice and has a unique tangent line (derivative) at every point of the interval, then somewhere between the endpoints it has a tangent parallel to the x-axis. The theorem was proved in 1691 by the French mathematician Michel Rolle, though it was stated without a modern formal proof in the 12th century by the Indian mathematician Bhaskara II.

#### Rolle’s theorem :Proof

Statement: If f(x) is

(i) continuous in the closed interval [a, b];

(ii) differentiable in the open interval (a, b).

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

Then, there exists at least one number \(c\in (a, b)\) such that f '(c) = 0.

If f(x) is constant then

f'(x) (c) = 0 for all c∈(a, b)

Suppose there exists x ∈ (a, b) such that

f(x) > f(a)

(A similar argument can be given if f(x) < f(a)).

Then there exists c ∈ (a, b) such that f(c) is a maximum.

Since, f(x) has a local maximum at c ∈(a, b), For small (enough) h,

\(f(c + h) \le f(c)\).

If h > 0 then

\( \frac{f(c + h)-f(c)}{h} \le 0\).

Similarly,

If h < 0, then

\(\frac{f(c + h)-f(c)}{h} \ge 0\).

By elementary properties of the limit, it follows that

f'(c) = 0.

#### Rolle’s theorem: Solved Examples

- Veryfy that the function satisfy the three hypothesis of Rolle's theorem, and find all numbers c that satisfy the conclusion of Rolle's theorem
- \( f(x)=x^2+2x\) over [-2,0]
- \( f(x)=x^3-4x\) over [-2,2]
- \( f(x)=x^2-4x+1\) over [0,4]
- \( f(x)=x^3-3x^2+2x+5\) over [0,2]
- \( f(x)=x^{2/3}\) over [0,1]
- \( f(x)=x^{2/3}\) over [-1,8]
- \( f(x)=x+\frac{1}{x}\) over [0.5,2]
- \( f(x)=\sin 2 x\) over [-1,1]
- \( f(x)=x \sqrt{x+6}\) over [-6,0]
- \( f(x)= \sqrt{x-1}\) over [1,3]

- For what value of a,m, and b does the function satisfy Roll's theorem in the interval [0,2]

\( f(x)= \begin{cases} 3 & x =0\\ -x^2+3x+a & 0 < x < 1\\ mx+b &1 \le x \le 2 \end{cases} \)

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