Exercises: Properties of differentiable functions. Rolle's, Mean Value and Cauchy's theorems

Given the polynomial function $f(x) = x^2 - 6x + 8$, verify whether the hypotheses of Rolle's Theorem are satisfied on the interval $[2, 4]$. If so, find the value of $c \in (2, 4)$ whose existence is guaranteed by the theorem.

Consider the function $f(x) = x^2 - kx + 10$. Determine the value of the parameter $k \in \mathbb{R}$ so that the function satisfies the condition $f(1) = f(5)$ and Rolle's Theorem can be applied on the interval $[1, 5]$. Once $k$ is found, find the value of $c$ where the derivative is zero.

Apply the Mean Value Theorem to the function $f(x) = x^2 - x$ on the interval $[0, 2]$. Find the value of $c$ whose existence is guaranteed by the theorem and explain graphically the relationship between the tangent line at $c$ and the secant line passing through the endpoints of the interval.

A marathon runner trains on a straight track. His or her position (in feet) relative to the starting point is given by the function $s(t) = t^2 + 2t$, where $t$ is the time in seconds. Use the Mean Value Theorem to prove that at some instant between $t=1$ and $t=4$ seconds, the runner's instantaneous velocity was exactly equal to the average velocity in that interval. Find this time.

Explain with reasoning why Rolle's Theorem cannot be applied to the function $f(x) = 1 - |x-1|$ on the interval $[0, 2]$, even though $f(0) = f(2) = 0$. Which hypothesis fails in this case?

Given the functions $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, verify whether the conditions of Cauchy's Theorem are satisfied on the interval $[1, 3]$. If so, find the value of $c \in (1, 3)$ such that:

It is known that a function $f(x)$ is continuous on the closed interval $[1, 5]$ and differentiable on the open interval $(1, 5)$. If the values of the function at the endpoints are $f(1) = 10$ and $f(5) = 30$, can we ensure that there is a point $c$ in the interval where the slope of the tangent line is equal to $5$? Justify your answer using the Mean Value Theorem.

Given the function $f(x) = \sin(x)$, verify that the hypotheses of Rolle's Theorem are satisfied on the interval $[0, \pi]$. Find the value of $c$ that satisfies the conclusion of the theorem and represent it schematically on a unit circle or on the graph of the function.

Given the piecewise function:

Prove that the hypotheses of the Mean Value Theorem are satisfied on the interval $[1, e+2]$ and find the value or values of $c \in (1, e+2)$ whose existence is guaranteed by the theorem.

An average speed camera on a highway detects that a vehicle enters the Lincoln Tunnel (point $A$) at 2:00:00 p.m. and exits it (point $B$) at 2:01:30 p.m. The distance between both control points is 3.5 miles.

If the speed limit in the tunnel is $120 \text{ mph}$, use the Mean Value Theorem to justify whether the driver should receive a speeding fine, regardless of whether their speed at the entrance and exit was legal.

Consider the polynomial function $f(x) = x^3 + x^2 - x + k$.

1. a) Determine the value of the parameter $k$ so that the function satisfies the hypotheses of Rolle's Theorem on the interval $[-2, 1]$.
2. b) For this value of $k$, find the points $c$ that satisfy the conclusion of the theorem.

Let the function be $f(x) = x \cdot e^{x^2-1}$.

1. a) Verify that $f(x)$ satisfies the hypotheses of the Mean Value Theorem on the interval $[0, 1]$.
2. b) Is there any point on this interval where the slope of the tangent line is equal to $e-1$? Justify your answer using the theorem.

Apply Cauchy's Theorem to the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ on the interval $\left[ 0, \frac{\pi}{2} \right]$. Find the value of $c$ that satisfies the equation:

Explain the geometric interpretation of this result in terms of the parametric curve defined by both functions.

Use Bolzano's Theorem and Rolle's Theorem to prove that the equation:

has exactly one real solution on the interval $[0, 1]$.

Consider the function $f(x) = \sqrt{x^2 + 9}$.

Find a point on the graph of the function where the tangent line is parallel to the secant line passing through the points on the curve with x-coordinates $x=0$ and $x=4$. Which theorem guarantees that this point exists? State its hypotheses and verify that they are satisfied.

Given the function $f(x) = \frac{x^2 - 1}{x - 3}$, can Rolle's Theorem be applied on the interval $[1, 5]$ knowing that $f(1) = f(5) = 0$?

Justify your answer by analyzing the continuity and differentiability of the function on this interval. If not, explain whether this contradicts the theorem.

Let the piecewise function be:

Find the values of the parameters $a, b, c \in \mathbb{R}$ so that the function $f(x)$ satisfies the hypotheses of Rolle's Theorem on the interval $[0, e]$. Once the parameters are found, determine the value of $c \in (0, e)$ whose existence is guaranteed by the theorem.

Prove, using both Bolzano's Theorem and Rolle's Theorem, that the equation:

has a single real solution for any value of the constant $k \in \mathbb{R}$. Explain with reasoning why the existence of more than one root would contradict the statement of Rolle's Theorem.

A speed control system (average speed cameras) is located on a highway between mile markers 150 and 162. A vehicle enters the section at 10:00:00 and leaves at 10:06:40.

1. a) Using the Mean Value Theorem, justify whether the vehicle has exceeded the speed limit of $100 \text{ mph}$ at any time.
2. b) If the function describing the position of the vehicle in that section is $s(t) = 150 + 2t - 0.03t^2$ (where $s$ is in miles and $t$ in minutes since entering the section), find the exact time when the instantaneous velocity coincides with the average velocity of the journey.

Given the functions $f(x) = e^x$ and $g(x) = x^2 + 1$, apply Cauchy's Mean Value Theorem (Generalized Mean Value Theorem) on the interval $[0, 2]$.

Find the value of $c \in (0, 2)$, corresponding to point $P$ in the figure, that satisfies the conclusion of the theorem:

Consider a function $f(x)$ differentiable on all of $\mathbb{R}$. It is known that the equation $f'(x) = 0$ has exactly two real solutions, $x_1 = -2$ and $x_2 = 3$.

Prove that the function $f(x)$ cannot have more than three distinct real roots. Is it possible for the function to have only one real root? Justify your answer based on the geometric interpretations of the theorems of differentiability.

Use the Mean Value Theorem to prove the following inequality for any pair of real numbers $a$ and $b$ such that $0 < a < b$:

Hint: Apply the theorem to the function $f(x) = \ln(x)$ on the interval $[a, b]$.

Consider the function $f(x) = x^3 - 3x + m$.

1. a) Determine for what values of the parameter $m$ the function satisfies the hypotheses of Rolle's Theorem on an interval of the form $[0, a]$ with $a > 0$.
2. b) For $m = 0$, find the largest possible interval $[p, q]$ where the theorem can be applied and find the point $c$ whose existence is guaranteed.

Two runners, Ashley and Brandon, start a 100-meter hurdles race at the same time and reach the finish line in a dead heat (same time $T$). Their position functions are $s_A(t)$ and $s_B(t)$, both continuous on $[0, T]$ and differentiable on $(0, T)$.

Prove, using Rolle's Theorem on the difference function $h(t) = s_A(t) - s_B(t)$, that there was at least one instant during the race when both runners had exactly the same instantaneous velocity.