Properties of Differentiable Functions. Rolle's, Mean Value, and Cauchy's Theorems

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

Explica razonadamente por qué no se puede aplicar el Teorema de Rolle a la función $f(x) = 1 - |x-1|$ en el intervalo $[0, 2]$, a pesar de que $f(0) = f(2) = 0$. ¿Qué hipótesis falla en este caso?

Given the function $f(x) = \frac{x^2 - 6x + 5}{x - 3}$, can Rolle's Theorem be applied on the interval $[1, 5]$ knowing that $f(1) = f(5) = 0$?

Justify your answer by analysing the continuity and differentiability of the function on that interval. If not, explain whether this contradicts 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 $k$ has been found, calculate the value of $c$ where the derivative is zero.

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

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

Let the piecewise function be defined as:

$f(x) = \begin{cases} x^2 + ax + b & \text{if } x \le 1 \\ k \ln(x) & \text{if } x > 1 \end{cases}$

Calculate the values of the parameters $a, b, k \in \mathbb{R}$ 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 the point $c \in (0, e)$ whose existence is guaranteed by the theorem.

Consider the function $f(x) = x^2 - x$ on the interval $[0, 2]$. The figure shows its graph and the secant line through the points $A$ and $B$ of the graph with $x$-coordinates $0$ and $2$.

1. a) Check that the hypotheses of the Mean Value Theorem (Lagrange) hold on $[0, 2]$ and calculate the slope of the secant line $AB$.

1. b) Find the value of the point $c \in (0, 2)$ whose existence is guaranteed by the theorem.

1. c) Explain graphically how the tangent line to the graph at the point with $x$-coordinate $c$ is related to the secant line $AB$.

A marathon runner trains on a straight track. Their position (in meters) 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 show that at some instant between $t=1$ and $t=4$ seconds, their instantaneous velocity was exactly equal to their average velocity on that interval. Calculate that instant.

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 guarantee that there exists a point $c$ in the interval where the gradient of the tangent line is equal to $5$? Justify your answer using the Mean Value Theorem.

Consider the function $f(x) = x \times e^{x^2-1}$.

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 in that interval where the slope of the tangent line is equal to $1$? Justify your answer using the theorem.

Given the piecewise function:

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

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 joining the points on the curve with x-coordinates $x=0$ and $x=4$. Which theorem guarantees that such a point exists? State its hypotheses and check that they are satisfied.

The GPS logging system of a freight train shows that it passed the 150 km mark of a railroad line at 10:00:00 and the 162 km mark at 10:06:40. The speed limit on that line is $100 \text{ km/h}$.

1. a) Using the Mean Value Theorem (Lagrange), justify whether the train exceeded the speed limit on the line at any point.
2. b) If the function describing the train's position on that section is $s(t) = 150 + 2t - 0.03t^2$ (where $s$ is in km and $t$ in minutes since passing the first kilometer mark), find the exact instant at which the instantaneous velocity coincides with the average speed of the trip.

Given the functions $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, check 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:

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

Explain the geometric interpretation of this result in terms of the parametric curve $t \mapsto (g(t), f(t))$, with $t \in \left[0, \frac{\pi}{2}\right]$. The figure shows that curve, its endpoints $A = (g(a), f(a))$ and $B = (g(b), f(b))$, the secant line $AB$, and the increments $g(b)-g(a)$ and $f(b)-f(a)$.

Given the functions $f(x) = e^x$ and $g(x) = x^2 + 1$, apply Cauchy's 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:

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

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

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

has a unique real solution for any value of the constant $k \in \mathbb{R}$. Give a reasoned explanation of why the existence of more than one root would contradict the statement of Rolle's 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 differentiability theorems.

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 which 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 calculate the point $c$ whose existence is guaranteed.

Two runners, Alba and Bruno, start a 100-meter hurdles race at the same time and reach the finish line in a photo finish (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.