question 17 of 19 this quiz: 95 point(s) possible this question: 5 point(s) possible determine any relative…

question 17 of 19 this quiz: 95 point(s) possible this question: 5 point(s) possible determine any relative maxima or minima of the function and the intervals on which the function is increasing or decreasing. f(x)= -x² + 9x - 14 does the function have a relative maximum or minimum? relative minimum relative maximum the relative maximum occurs at x = and has a value of on what interval is the function increasing? a. (6.25,∞) b. (4.5,∞) c. (-∞,6.25) d. (-∞,4.5) on what interval is the function decreasing? a. (4.5,∞) b. (6.25,∞) c. (-∞,4.5) d. (-∞,6.25)

question 17 of 19 this quiz: 95 point(s) possible this question: 5 point(s) possible determine any relative maxima or minima of the function and the intervals on which the function is increasing or decreasing. f(x)= -x² + 9x - 14 does the function have a relative maximum or minimum? relative minimum relative maximum the relative maximum occurs at x = and has a value of on what interval is the function increasing? a. (6.25,∞) b. (4.5,∞) c. (-∞,6.25) d. (-∞,4.5) on what interval is the function decreasing? a. (4.5,∞) b. (6.25,∞) c. (-∞,4.5) d. (-∞,6.25)

Answer

Explanation:

Step1: Identify the function type

The function $f(x)=-x^{2}+9x - 14$ is a quadratic function of the form $y = ax^{2}+bx + c$ where $a=-1$, $b = 9$, $c=-14$.

Step2: Find the x - coordinate of the vertex

The x - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. Substituting $a=-1$ and $b = 9$ into the formula, we get $x=-\frac{9}{2\times(-1)}=\frac{9}{2}=4.5$.

Step3: Determine if it's a max or min

Since $a=-1<0$, the parabola opens downwards, so the function has a relative maximum.

Step4: Find the y - coordinate of the vertex

Substitute $x = 4.5$ into the function $f(x)=-x^{2}+9x - 14$. So $f(4.5)=-(4.5)^{2}+9\times4.5-14=-20.25 + 40.5-14=6.25$.

Step5: Determine increasing and decreasing intervals

For a parabola that opens downwards, the function is increasing on the interval $(-\infty,x_{vertex})$ and decreasing on the interval $(x_{vertex},\infty)$. So it is increasing on $(-\infty,4.5)$ and decreasing on $(4.5,\infty)$.

Answer:

Does the function have a relative maximum or minimum? Relative maximum The relative maximum occurs at $x = 4.5$ and has a value of $6.25$. On what interval is the function increasing? D. $(-\infty,4.5)$ On what interval is the function decreasing? A. $(4.5,\infty)$