1. determine el dominio máximo de la siguiente función $f$:\n$$f(x) = \\sqrt{\\frac{x^3 + 4x^2 - 17x…

Explanation:

Step1: Identify domain constraints

The radicand must be non-negative and the denominator non-zero. $$ \frac{x^3 + 4x^2 - 17x - 60}{2x^3 + 3x^2 - 18x + 8} \geq 0 $$

Step2: Factor the numerator

Find roots of $x^3 + 4x^2 - 17x - 60$ using synthetic division. $$ (x - 4)(x + 3)(x + 5) $$

Step3: Factor the denominator

Find roots of $2x^3 + 3x^2 - 18x + 8$ using synthetic division. $$ (x - 2)(x + 4)(2x - 1) $$

Step4: Set up the inequality

Analyze the signs of the factored rational expression. $$ \frac{(x - 4)(x + 3)(x + 5)}{(x - 2)(x + 4)(2x - 1)} \geq 0 $$

Step5: Determine critical points

Identify roots and vertical asymptotes in ascending order. $$ x \in {-5, -4, -3, 0.5, 2, 4} $$

Step6: Perform sign analysis

Test intervals between critical points to satisfy the inequality. $$ (-\infty, -5]: +, (-5, -4): -, (-4, -3]: +, (-3, 0.5): -, (0.5, 2): +, [4, \infty): + $$

Step7: Exclude denominator zeros

Exclude $x = -4$, $x = 0.5$, and $x = 2$ from the domain. $$ x \in (-\infty, -5] \cup (-4, -3] \cup (0.5, 2) \cup [4, \infty) $$

Answer:

$(-\infty, -5] \cup (-4, -3] \cup \left(\frac{1}{2}, 2\right) \cup [4, \infty)$