Question

suppose y = f(x) graphed in the figure below represents the cost of manufacturing x kilograms of a chemical. f(x)/x represents the average cost of producing 1 kilogram of the chemical when x kilograms are made. this problem asks you to visualize these averages graphically. (a) we can represent f(4)/4 as the slope of a line. through what points does this line extend? ( ) and ( ) (enter your answers as (x,y) coordinates, where the x - coordinate in each case is an integer value and the y - coordinate is either an integer or the y - value on the curve.) (b) complete the inequality: f(3)/3 symbol f(4)/4

Explanation:

Step1: Recall slope - formula

The slope of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. We know that $\frac{f(4)}{4}=\frac{f(4)-0}{4 - 0}$.

Step2: Identify the points for $\frac{f(4)}{4}$

For the slope $\frac{f(4)}{4}$, the line passes through the origin $(0,0)$ and the point $(4,f(4))$ on the curve $y = f(x)$.

Step3: Analyze the slopes for comparison

The slope of the line from $(0,0)$ to $(3,f(3))$ is $\frac{f(3)}{3}$, and the slope of the line from $(0,0)$ to $(4,f(4))$ is $\frac{f(4)}{4}$. Looking at the graph of $y = f(x)$ (a convex - up curve), the slope of the line from the origin to a point on the curve increases as $x$ increases. So, $\frac{f(3)}{3}<\frac{f(4)}{4}$.

Answer:

(a) $(0,0)$ and $(4,f(4))$
(b) $<$