Rolle's Theorem Verification: Calculus Problems and Solutions

Published on July 10, 2025

Calculus

Verify Rolle's Theorem for two functions: f(x) = e^x(sin x - cos x) and f(x) = x^3 - 6x^2 + 11x - 6, by checking the conditions of the theorem and finding the value(s) of c where the derivative is zero.

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Answer

The user wants to solve a math problem related to Rolle’s Theorem. The problem has two parts.

Part 1: Verify Rolle’s Theorem for in .

Rolle’s Theorem states that if a function

satisfies the following three conditions:

is continuous on the closed interval .
is differentiable on the open interval .
.
Then there exists at least one number in such that .

Let’s check these conditions for

Condition 1: Continuity
The functions

, and

are continuous everywhere.
The product of continuous functions is continuous, and the difference of continuous functions is continuous.
Therefore,

is continuous on

Condition 2: Differentiability
We need to find the derivative

.
Using the product rule

, where

and

So,

Since

and

are differentiable everywhere,

exists for all

.
Therefore,

is differentiable on

Condition 3:
Here

and

We know that

and

Since

and

, we have

All three conditions of Rolle’s Theorem are satisfied. Therefore, there exists at least one

such that

Let’s find

Since

for all real

, we must have

, which implies

.
In the interval

, the value of

for which

.
Since

and

, and

, we have

Thus, Rolle’s Theorem is verified for the given function and interval.

Part 2: Verify Rolle’s Theorem for in .

Let’s check the conditions for

Condition 1: Continuity

is a polynomial function. All polynomial functions are continuous everywhere.
Therefore,

is continuous on

Condition 2: Differentiability
We need to find the derivative

Since

is a polynomial, it exists for all

.
Therefore,

is differentiable on

Condition 3:
Here

and

Since

and

, we have

All three conditions of Rolle’s Theorem are satisfied. Therefore, there exists at least one

such that

Let’s find

This is a quadratic equation. We can use the quadratic formula

.
Here

We have two possible values for

and

Let’s check if these values are in the interval

Since

is in the interval.

Since

is in the interval.

Both values of

are in the interval

. Thus, Rolle’s Theorem is verified for the given function and interval.

The final answer is

For the first function,

is continuous on because , , and are continuous everywhere.
exists for all , so is differentiable on .
.
.
So, .
All conditions are met. Setting : . The value is in .

For the second function,

is a polynomial, so it is continuous on .
exists for all , so is differentiable on .
.
.
So, .
All conditions are met. Setting : .
Using the quadratic formula, .
Both and are in the interval .

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Rolle's Theorem Verification: Calculus Problems and Solutions

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