Question

question 5 (4 points)
use desmos to find the turning point for the function
$f(x) = \frac{5}{x^2 + 2} + 3.5$
options: (5, 2); (2, 5); (2, 3.5); (0, 6)
question 6 (4 points)
suppose $f(x) = x^2 - 4x$ million dollars are earned, where $x$ is the number of years after 2000. what should $x$ be to represent the year 2010?
options: 4032060; 10; 60; 2010
question 7 (4 points)
find the slope of the line containing both of the following points:
$(2, -5)$ and $(4, -11)$
options: $\frac{1}{3}$; $-3$; $3$; $-\frac{1}{3}$

Explanation:

Question 5

Step1: Identify critical point condition

To find the turning point, take the derivative of $f(x)$ and set it to 0:
$f'(x) = -\frac{10x}{(x^2+2)^2} = 0$

Step2: Solve for x-value

The numerator must be 0: $-10x=0 \implies x=0$

Step3: Find corresponding y-value

Substitute $x=0$ into $f(x)$:
$f(0)=\frac{5}{0^2+2}+3.5 = 2.5+3.5=6$

Question 6

Step1: Calculate years after 2000

Subtract 2000 from 2010:
$x=2010-2000$

Step2: Compute the result

$x=10$

Question 7

Step1: Use slope formula

Slope $m=\frac{y_2-y_1}{x_2-x_1}$ for points $(x_1,y_1)=(2,-5)$ and $(x_2,y_2)=(4,-11)$

Step2: Substitute values and calculate

$m=\frac{-11-(-5)}{4-2}=\frac{-6}{2}=-3$

Answer:

Question 5: (0, 6)
Question 6: 10
Question 7: $-3$