The image contains a series of mathematical exercises designed for high school level, likely covering topics such as propositional logic, function analysis, and calculus. I'll break down each exercise and provide a general approach or solution outline.

Exercise 1

This exercise focuses on logic, equations, and recurrence relations.

1. Negation and Truth Value of Proposition P:

   • Given $P: \forall x \in \mathbb{R}, \ x^2 = x$, the goal is to determine the negation and truth value of this proposition.

2. Prove a Statement Involving Squares:

   • Show that $(\forall x \in \mathbb{R}) \ x^2 \leq x + 1$.

3. Solve an Equation:

   • Solve $6x - 3 = 2x + 2$ in $\mathbb{R}$.

4. Inequality Proof:

   • Show that $\forall (a \in \mathbb{R}) \ \exists (b \in \mathbb{R}) : a^2 - 2a + 1 \leq 2ab$.

5. Recurrence Relation Proof:

   • Prove by recurrence that $\forall n \in \mathbb{N}, \ 1 + 6 + 6^2 + \dots + 6^n = \frac{6^{n+1} - 1}{5}$.

6. Value of Expression:

   • Deduce the value of the expression $A = 1 + 6 + 6^2 + \dots + 6^6$.

Exercise 2

This exercise involves analyzing two functions $f$ and $g$.

• Functions:
  • $f(x) = \frac{2}{\sqrt{x}} + 3$
  • $g(x) = x^2 - 4x + 5$

Tasks:

1. Determine Domains:

   • Find the domains of $f$ and $g$.

2. Variation Tables:

   • Draw the variation tables for $f$ and $g$.

3. Graph Sketching:

   • Sketch the graphs of $f$ and $g$ on the same Cartesian plane in an orthonormal coordinate system.

4. Composite Function Domain:

   • Determine the domain of $g \circ f$.

5. Evaluate Composite Function:

   • Find $g(f(x))$ for $x \in D_{g \circ f}$.

6. Monotonicity of Composite Function:

   • Determine the monotonicity of $g(f(x))$ on intervals $[-3, 1]$ and $[1, +\infty]$.

Exercise 3

This exercise provides graphs for two functions $f$ and $g$ and involves graphical analysis.

1. Determine Specific Values:

   • Find $f(-3)$, $g(-1)$, $f(3)$, $g(1)$, $g(3)$, and $f(5)$.

2. Variation Table of $f$:

   • Draw the variation table of $f$.

3. Extrema of $g$:

   • Identify the extrema (maximum and minimum points) of $g$.

4. Graph Interpretation:

   • Solve graphically for:
     • $g(x) = f(x)$
     • $g(x) < 0$
     • $|g(x)| > |f(x)|$

Would you like a detailed solution for a specific exercise or task?

Related Questions

1. How do you determine the truth value of a proposition and its negation?
2. What is the process for solving recurrence relations?
3. How can you analyze and compare the monotonicity of composite functions?
4. How do variation tables help in understanding functions’ behaviors?
5. What methods can be used to solve inequalities graphically?

Tip

For analyzing functions, always start with finding their domain before attempting any further operations or compositions. This ensures the functions are defined for the values you're considering.