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The average rate of change on the interval (5, 6) is 9. So the correct option is D.
Here we want to find the average rate of change of f(x), the function on the table, on the interval (5, 6).
This is just:
[tex]r = \frac{f(6) - f(5)}{6 - 5}[/tex]
[tex]f(x) = a*x^2 + b*x + c[/tex]
I we look at the table we see that:
[tex]f(1) = 1 = a + b + c[/tex]
[tex]f(2) =2 = 4a + 2b + c[/tex]
[tex]f(3) = 5 = 9a + 3b + c[/tex]
This is a system of equations.
If we subtract the second and first functions, we get:
[tex]2 - 1 = (4a + 2b +c) - (a + b + c)\\1 = 3a + b = a + b + c[/tex]
From that we take two relations:
[tex]1 - 3a = b\\2a = c[/tex]
Now we can replace these two in the last equations so we get:
[tex]5 = 9a + 3b + c\\\\5 = 9a + 3*(1 - 3a) + 2a\\\\5 = 9a + 3 - 9a + 2a\\\\5 = 3 + 2a\\\\5 - 3 = 2a\\\\2 = 2a\\\\a = 1[/tex]
Now that we know the value of a:
[tex]c = 2a = 2*1 = 2\\\\b = 1 - 3a = 1 - 3 = -2[/tex]
The quadratic equation is:
[tex]f(x) = x^2 - 2x + 2[/tex]
Evaluating this in x = 6 we get:
[tex]f(6) = 6^2 - 2*6 + 2 = 26[/tex]
And from the table we know that f(5) = 17, then the average rate of change is:
[tex]r = \frac{f(6) - f(5)}{6 - 5} = \frac{26-17}{1} = 9[/tex]
The correct option is D.
If you want to learn more about average rates of change:
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