Let f(x) = 1/x . Find the number b such that the average rate of change of f on the interval [2, b] is − 1/8

We want to find an interval such that the given equation, f(x) = 1/x, has an average rate of change of -1/8 in that interval.

We will see that the interval is [2, 4]

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For a function f(x), the average rate of change in the interval [a, b] is given by:

Here we have:

And the interval is [2, b] such that r in that interval is -1/8, so we need to solve:

We can rewrite it to:

Now we can remove the term (b - 2) because it appears on both sides, so we get:

Then we found that b must be equal to 4, so the interval is [2, 4]

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b=4

Step-by-step explanation:

So, we have the function . We need to find b such that the average rate of change or the slope is -1/8 between the intervel [2, b]. First, let's find f(2).

f(2) = 1/(2) = 1/2

So, we have the point (2, 1/2)

At point b, f(b) = 1/b.

Let's plug this into the slope formula:

Now, we just need to solve for b. First, let's multiply both the numerator and denominator by b (to get rid of the annoying fraction in the numerator).

Now, cross multiply.

Solve for b. Factor using the numbers -4 and -2.

Thus, b=4 or b=2.

However, b=2 is not a possible solution since the interval [2,2] means nothing. Thus, b=4.

step-by-step explanation:

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