Which two points in the graphed function have an average rate of change of 5?

B. points A and B

Step-by-step explanation:

"Rate of change" is just a fancy name for slope.

The question asks for points on the graph having a slope of 5.  We know a slope of 5 is a positive (goes up from left to right) and it's quite steep.

By looking at the graph, we see that to have a slope of 5, it has to take place in the first half of the graph.

Let's look at the possible options:

A. Points D and F: From point D to point F, you're going down, so the slope is negative.  NO

B. Points A and B: goes up, pretty steep.  Let's calculate the slope.

Point A: (2,1), Point B: (3,6)

Slope = (6 - 1) / (3 - 2) = 5 / 1 = 5

We found it!

C. Points B and C: goes up, pretty steep too.  Let's calculate the slope:

Point B: (3,6) , Point C: (4,9)

Slope = (9 - 6) / (4 - 3) = 3 / 1 = 3, NO, not the slope we're looking for

D. Points C and E: goes down, NO,  not what we want.

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.