Graph the linear function p(x)=4x .

Please find attached the required graph

Step-by-step explanation:

The given equation of the linear function is p(x) = 4·x

Comparing the above equation with the general equation for a straight line, y = m·x + c, we have;

The slope, m = 4, and the y-intercept c = 0

Therefore, the graph can be constructed by drawing a straight line passing through the origin which is inclined to the horizontal axis such that the ratio of the vertical distance of a point on the line from the x-axis to the distance of the point from the y-axis is 4 to 1

However, in order to  graph the given function, such that the values of the x and y coordinates of the points on the line representing the equation of the graph are clearly defined, we generate a table of values  using Microsoft Excel as follows;

x                            p(x)

0                            0

1                             4

2                            8

3                            12

4                            16

5                            20

6                            24

7                            28

8                            32

9                            36

10                            40

11                            44

12                            48

13                            52

14                            56

15                            60

16                            64

From which the required graph is constructed using the insert function in Microsoft Excel

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.