Respuesta :
Answer:
Step-by-step explanation:
Given the function \( f(x) = [tex]x^3 - 10x^2 + 27x[/tex] - 18 \), we want to find its key characteristics:
1. Domain:
The domain of a polynomial function is all real numbers, so for \( f(x) \), the domain is [tex]\( (-\infty, +\infty) \).[/tex]
2. Range:
Since[tex]\( f(x) \)[/tex] is a cubic polynomial, its range is also all real numbers \( [tex](-\infty, +\infty) \).[/tex]
3. Relative Maximum(s) and Minimum(s):
To find relative maxima and minima, we need to find critical points and use the second derivative test or the first derivative test.
[tex]The first step is to find the derivative of \( f(x) \):\[ f'(x) = 3x^2 - 20x + 27 \]To find critical points, set \( f'(x) = 0 \):\[ 3x^2 - 20x + 27 = 0 \][/tex]
Using the quadratic formula, we find the roots:
[tex]\[ x = \frac{20 \pm \sqrt{400 - 4(3)(27)}}{6} \]\[ x = \frac{20 \pm \sqrt{400 - 324}}{6} \]\[ x = \frac{20 \pm \sqrt{76}}{6} \]\[ x = \frac{20 \pm 2\sqrt{19}}{6} \]\[ x = \frac{10 \pm \sqrt{19}}{3}[/tex]
4. End Behavior:
As [tex]\( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \) and as \( x \rightarrow +\infty \), \( f(x) \rightarrow +\infty \).[/tex]
5. Increase and Decrease Intervals:
To find intervals of increase and decrease, we need to analyze the sign of the first derivative.
6. Zeros:
The zeros of the function are the values of \( x \) for which \( f(x) = 0 \).
After calculating the critical points, you can use them to find intervals of increase and decrease. Finally, use the sign of the second derivative to determine the nature of the critical points as relative maxima or minima.
Once you find these characteristics, you can summarize them for the function( f(x) = [tex]x^3 - 10x^2 + 27x[/tex] - 18 ).
