For which function is the average rate of change over the interval 1 < x < 5 greater than the average rate of change over the same interval for the function g (x)

We do not have the function g (x), but this can be solved in a trivial way.

The average rate of change in a interval is the slope over that interval, so the average rate of change of g (x) in the interval 1 < x < 5 is

p = (g (x2) - g (x1)) / (x2 - x1) where x2 > x1

p = (g (5) - g (1)) / (5 - 1) = (g (5) - g (1)) / 4

Now, suppose that i have the new equation h (x) = g (x) + x

the rate of change of this function, the average rate of change in the interval will be:

p' = (h (5) - h (1)) / (5 - 1) = (g (5) + 5 - g (1) - 1) / 4 = (g (5) - g (1)) / 4 + 4/4 = p + 1

so p' > p

and the average rate of change of h (x) is bigger than the one of g (x) in the interval 1< x < 5

Answer: x=2 and x=6