Place the graphs in descending order based on their y value as x approaches infinity.

To answer this question, we need to know that as x tends to infinity

the exponential values greatly increase with each passing x.

So the exponential values will always be greater than the linear values, as x increases.

Lets see that with an example

Suppose we have

y1 = 2*x

y2 =  2^x

If we evaluate at x = 5, we can see the difference between the two

y1 = 2*(5) = 10

y2 =  2^(5) = 32

If x = 10, the difference becomes even clearer

y1 = 2*(10) = 20

y2 =  2^(10) = 1024

To order the exponential graphs, we need to place ourselves at an specific value of x and check which graphs has the highest corresponding value of y

So the answer to your question, which is to place in descending order according to the value of y:

*green

*orange

*blue

*red

*yellow

*black

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.