Question

A differentiable function f has the property that f'(x) < 3 for 1 < x < 8 and f(5) = 6. Which of the following could be true?

1) f(2) = 0
2) f(6) = -2
3) f(7) = 13

Answer 1

Based on the Mean Value Theorem, options 1 (f(2) = 0) and 3 (f(7) = 13) could potentially be true for the differentiable function given the constraints on the derivative. Option 2 (f(6) = -2) can be ruled out immediately as it would require a rate of change outside the given conditions.

Step-by-step explanation:

The question asks us to determine which of the values could be true for a function f based on given conditions: f'(x) < 3 for 1 < x < 8 and f(5) = 6. To figure this out, we need to apply the concept of the Mean Value Theorem, which in simple terms, tells us that if a function is continuous and differentiable over a certain interval, there must be at least one point in the interval where the derivative of the function is equal to the average rate of change over that interval.

Let's consider each option in light of the given information:

1. f(2) = 0 could be true, since the average rate of change from x=2 to x=5 would be (6-0)/(5-2) = 2, which is less than 3.
2. f(6) = -2 could not be true, since the average rate of change from x=5 to x=6 would be (-2-6)/(6-5) = -8, which is not possible given that f'(x) is always less than 3.
3. f(7) = 13 could also be true, since the average rate of change from x=5 to x=7 would be (13-6)/(7-5) = 3.5. However, this is not immediately disqualified as we only know that f'(x) is less than 3, not necessarily equal to 3. Therefore, a derivative less than 3 over the interval is still consistent with the Mean Value Theorem as the actual derivative could be fluctuating but always staying below 3.

Hence, both options 1 and 3 could potentially be true given the conditions, but we can immediately rule out option 2.

Answer 2

Using the Mean Value Theorem, we can determine which statements are true about the function f(x). The only true statement is that f(6) = -2.

Step-by-step explanation:

To solve this problem, we need to use the Mean Value Theorem. The Mean Value Theorem states that if a function is continuous on a closed interval [a, b], and differentiable on the open interval (a, b), then there exists a number c in the open interval (a, b) such that f'(c) = ∆y/∆x, where ∆y is the change in y-values and ∆x is the change in x-values. In this case, we are given that f'(x) < 3 for 1 < x < 8 and f(5) = 6.

So, let's consider option 1) f(2) = 0. Since 1 < 2 < 8, we can apply the Mean Value Theorem to the interval (1, 2). If f(2) = 0, then f(1) = 6 - ((2-5)/(5-1)) * 3 = 0, which is not consistent with the given information. Therefore, option 1) is not true.

Now, let's consider option 2) f(6) = -2. Since 1 < 6 < 8, we can apply the Mean Value Theorem to the interval (1, 6). If f(6) = -2, then f(5) = 6 - ((6-5)/(5-1)) * 3 = -2, which is consistent with the given information. Therefore, option 2) is true.

Finally, let's consider option 3) f(7) = 13. Since 1 < 7 < 8, we can apply the Mean Value Theorem to the interval (1, 7). If f(7) = 13, then f(5) = 6 - ((7-5)/(5-1)) * 3 = 0, which is not consistent with the given information. Therefore, option 3) is not true.

In conclusion, the only true statement is that f(6) = -2, option 2).