Properties of differentiable functions. Rolle's theorem, the mean value theorem and Cauchy's theorem

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.

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.

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.

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 $f(x) = x^2 - x$ 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 metres) 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.

An average speed camera on an Irish motorway detects that a vehicle enters the Dublin Port Tunnel (point $A$) at 14:00:00 and leaves it (point $B$) at 14:01:30. The distance between both control points is 3.5 km.

If the speed limit in the tunnel is $120 \text{ km/h}$, use the Lagrange 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.

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 $f(b)-f(a)$.

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

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

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 $[a, b]$.

It is desired to prove rigorously that the equation $2x^3 + 3x - 4 = 0$ has exactly one real solution in the open interval $(0, 2)$. To tackle this problem, two students propose different approaches:

• Strategy A (Bolzano and Monotonicity): Prove the existence of at least one root in $(0, 2)$ using Bolzano's Theorem and then prove that the associated function is strictly increasing on all of $\mathbb{R}$ by analysing the sign of $f'(x)$ to deduce uniqueness.

• Strategy B (Bolzano and Rolle's Theorem): Prove the existence of at least one root with Bolzano and then assume by contradiction that there exist two distinct roots $x_1 < x_2$ in $(0, 2)$, applying Rolle's Theorem on $[x_1, x_2]$ to reach a contradiction with the equation $f'(x) = 0$.

You are asked to:

1. a) Critically compare both strategies: argue why both are valid and explain which of the two is methodologically more direct when the first derivative has no real roots.
2. b) Choose one of the two strategies and write out the complete, rigorous proof of the existence and uniqueness of the root, explicitly verifying all required hypotheses of the theorems you use.