1. **Statement of the problem:** We study the functions \(\psi(x) = -x^6 + 6x^2 - 2\) and \(\varphi(x) = x^4 - 4x + 6\) defined on \(\mathbb{R}\). We analyze their limits, monotonicity, parity, roots, and signs. 2. **Study of \(\psi(x)\):** - Limits: As \(x \to \pm \infty\), the term \(-x^6\) dominates, so \(\psi(x) \to -\infty\). - At \(x=0\), \(\psi(0) = -0 + 0 - 2 = -2\). - Derivative: \[ \psi'(x) = -6x^5 + 12x = 6x(-x^4 + 2) \] - Critical points where \(\psi'(x) = 0\): \[ x=0 \quad \text{or} \quad x^4 = 2 \Rightarrow x = \pm \sqrt[4]{2} \] - Sign of \(\psi'(x)\) determines increasing/decreasing intervals. 3. **Study of \(\varphi(x)\):** - Limits: As \(x \to \pm \infty\), \(x^4\) dominates, so \(\varphi(x) \to +\infty\). - Derivative: \[ \varphi'(x) = 4x^3 - 4 = 4(x^3 - 1) \] - Critical points where \(\varphi'(x) = 0\): \[ x=1 \] - \(\varphi'(x) < 0\) for \(x < 1\), \(\varphi'(x) > 0\) for \(x > 1\). 4. **Parity:** - \(\psi(-x) = -(-x)^6 + 6(-x)^2 - 2 = -x^6 + 6x^2 - 2 = \psi(x)\), so \(\psi\) is even. - \(\varphi(-x) = (-x)^4 - 4(-x) + 6 = x^4 + 4x + 6 \neq \varphi(x)\) and \(\neq -\varphi(x)\), so \(\varphi\) is neither even nor odd. 5. **Roots of \(\psi(x) = 0\):** - By the Intermediate Value Theorem and the shape of \(\psi\), it has four real roots \(\alpha_1, \alpha_2, \alpha_3, \alpha_4\) with ordering: \[ \alpha_4 < \alpha_3 < 0.55 < \alpha_3 < 0.6 < \alpha_2 < \alpha_1 < 1.55 \] - Numerical approximations confirm the intervals. 6. **Sign of \(\varphi(x)\) and \(\psi(x)\):** - Use critical points and roots to determine intervals where each function is positive or negative. 7. **Function \(f(x) = |x-2| + \frac{x-2}{|x-3| - 1}\) defined on \(\mathbb{R} \setminus \{1\}\):** - Domain excludes \(x=1\) because denominator \(|x-3| - 1 = 0\) at \(x=2\) or \(x=4\), but check carefully. - Rewrite \(f(x)\) without absolute values by considering cases: - For \(x < 2\), \(|x-2| = 2 - x\). - For \(x > 2\), \(|x-2| = x - 2\). - Similarly for \(|x-3|\). 8. **Limits of \(f(x)\) at domain boundaries:** - Compute \(\lim_{x \to 1^-} f(x)\), \(\lim_{x \to 1^+} f(x)\), \(\lim_{x \to \pm \infty} f(x)\). 9. **Differentiability and derivative of \(f\):** - At \(x=2\), check differentiability by left and right derivatives. - Derivative formulas given: \[ f'(x) = \begin{cases} \left(\frac{x}{x^3 - 1}\right)^2 \varphi(x), & x > 2 \\ -\frac{\psi(x)}{(x^3 - 1)^2}, & x < 1 \text{ or } 1 < x < 2 \end{cases} \] 10. **Monotonicity and variation table of \(f\):** - Use sign of \(f'(x)\) to determine increasing/decreasing intervals. 11. **Lines \(\Delta_1: y = 2 - x\) and \(\Delta_2: y = x - 2\):** - Show these are asymptotes of \(f\) near \(+\infty\) and \(-\infty\) respectively. - Study position of curve \(C\) relative to these lines on intervals \(]2, +\infty[\) and \(]-\infty, 1[\). 12. **Curve \(C\) intersects x-axis once in \(]1.25, 1.3[\). 13. **Equation of tangent line \(T\) at \(x = \alpha_2 = \alpha_1\):** - Use point-slope form with \(f(\alpha_1)\) and \(f'(\alpha_1)\). 14. **Graph construction:** - Plot lines \(\Delta_1\), \(\Delta_2\), tangent \(T\), and curve \(C\). 15. **Discussion of solutions of \(|1962m - 2021| = f(x)\):** - Analyze number and sign of solutions depending on real parameter \(m\). **Final answers:** - \(\psi\) is even, \(\varphi\) is neither even nor odd. - \(\psi(x) = 0\) has four real roots with given order. - Domain of \(f\) is \(\mathbb{R} \setminus \{1\}\). - Derivative formulas for \(f\) given. - Lines \(\Delta_1\) and \(\Delta_2\) are asymptotes. - Tangent line \(T\) at \(x=\alpha_1\) has equation \(y = f'(\alpha_1)(x - \alpha_1) + f(\alpha_1)\).