Determine if the function f(x) = x − x³ satisfies the conditions of Rolle's theorem on the interval [−1, 0] and [0, 1]. In the affirmative case, determine the values of c.

Exercise 3

Does the function f(x) = 1 − x satisfy the conditions of Rolle's theorem on the interval [−1, 1]?

Exercise 4

Prove that the equation 1 + 2x + 3x² + 4x³ = 0 has a unique solution.

Exercise 5

How many roots does the equation x³ + 6x² + 15x − 25 = 0 have?

Exercise 6

Prove that the equation 2x³ − 6x + 1 = 0 has only one real solution on the interval (0, 1).

Exercise 7

Can the mean value theorem be applied to f(x) = 4x² − 5x + 1 on [0, 2]?

Exercise 8

Can the mean value theorem be applied to f(x) = 1/ x² on [0, 2]?

Exercise 9

In the segment of the parabola between the points A = (1, 1) and B = (3, 0), find a point whose tangent is parallel to the chord.

Exercise 10

Calculate a point on the interval [1, 3] in which the tangent to the curve y = x³ − x² + 2 is parallel to the line determined by the points A = (1, 2) and B = (3, 20). What theorem guarantees the existence of this point?

Exercise 11

Determine a and b for the function:

If it satisfies the hypothesis of mean value theorem on the interval [2, 6].

Solution of exercise 1

Is Rolle's theorem applicable to the function f(x) = |x - 1| on the interval [0, 2]?

The function is continuous on [0, 2].

Rolle's Theorem is not applicable to the function because it is not differentiable at x = 1.

Solution of exercise 2

Determine if the function f(x) = x − x³ satisfies the conditions of Rolle's theorem on the interval [−1, 0] and [0, 1]. In the affirmative case, determine the values of c.

f(x) is a continuous function on the interval [−1, 0] and [0, 1], and differentiable on the open intervals (−1, 0) and (0, 1) because it is a polynomial function.

Also, it is determined that:

f(−1) = f(0) = f(1) = 0

Therefore, Rolle's Theorem is applicable.

Solution of exercise 3

Does the function f(x) = 1 − x satisfy the conditions of Rolle's Theorem on the interval [−1, 1]?

The function is continuous on the interval [−1, 1] and differentiable on (−1, 1) because it is a polynomial function.

Rolle's Theorem is not satisfied because f(−1) ≠ f(1).

Solution of exercise 4

Prove that the equation 1 + 2x + 3x² + 4x³ = 0 has a unique solution.

It can be proven by the reductio ad absurdum arguement.

If the function had two different roots x_{1} and x_{2}, being x_{1}< x_{2} , there would be:

f(x_{1}) = f(x_{2}) = 0

And since the function is continuous and differentiable (as it is a polynomial function), Rolle's theorem can be applied, which states that c
(x_{1}, x_{2}) exists such that f' (c) = 0.

f' (x) = 2 + 6x + 12x² f' (x) = 2 (1+ 3x + 6x²).

But f'(x) ≠ 0 and does not admit real solutions because the discriminant is negative:

Δ = 9 − 24 < 0.

Since the derivative is not annulled at any value, it contradicts Rolle´s theorem and it is determined that there are two false roots.

Solution of exercise 5

How many roots does the equation x³ + 6x² + 15x − 25 = 0 have?

The function f(x) = x³ + 6x² + 15x − 25 is continuous and differentiable on
·

So, the equation has at least one solution in the interval (0, 1).

Rolle's theorem.

f' (x) = 6x² - 6 6x² - 6 = 0 6(x − 1) (x + 1) = 0

The derivative is annulled at x = 1 and x = −1, therefore there cannot be two roots in the interval (0, 1).

Solution of exercise 7

Can the mean value theorem be applied to f(x) = 4x² − 5x + 1 on [0, 2]?

f(x) is continuous on [0, 2] and differentiable on (0, 2) therefore it is possible to apply the intermediate value theorem:

Solution of exercise 8

Can the mean value theorem be applied to f(x) = 1/ x² on [0, 2]?

The function is not continuous on [−1, 2] since is not defined at x = 0.

Solution of exercise 9

In the segment of the parabola between the points A = (1, 1) and B = (3, 0), find a point whose tangent is parallel to the chord.

The points A = (1, 1) and B = (3, 0) belong to the parabola of equation y = x² + bx + c.

Since the function is a polynomial, the mean value theorem can be applied on the interval [1, 3].

Solution of exercise 10

Calculate a point on the interval [1, 3] in which the tangent to the curve y = x³ − x² + 2 is parallel to the line determined by the points A = (1, 2) and B = (3, 20). What theorem guarantees the existence of this point?

Find the equation of the straight line that passes through the two points.

Since y = x³ − x² + 2 is continuous on [1, 3] and differentiable on (1, 3), the intermediate value theorem can be applied:

Solution of exercise 11

Determine a and b for the function:

If it satisfies the hypothesis of the mean value theorem on the interval [2, 6].

First, it must fulfill that the function is continuous on [2, 6].

Second, it must be determined that the function is differentiable on (2, 6).

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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Yeabsira Guest

23 Nov.

Thank you very much for spending your time preparing this article. It’s very helpful.

Guest

Thank you very much for spending your time preparing this article. It’s very helpful.