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.

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 a Spanish motorway detects that a vehicle enters the Guadarrama 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 $[a, b]$.