let f be a function that is continuous on the closed interval 1,3 with f(1)=10 and f(3)=18. which of the…

let f be a function that is continuous on the closed interval 1,3 with f(1)=10 and f(3)=18. which of the following statements must be true? a 10 ≤ f(2) ≤ 18 b f is increasing on the interval 1,3. c f(x)=17 has at least one solution in the interval 1,3. d f(x)=8 has at least one solution in the interval (1,3). e ∫₁³ f(x)dx > 20

let f be a function that is continuous on the closed interval 1,3 with f(1)=10 and f(3)=18. which of the following statements must be true? a 10 ≤ f(2) ≤ 18 b f is increasing on the interval 1,3. c f(x)=17 has at least one solution in the interval 1,3. d f(x)=8 has at least one solution in the interval (1,3). e ∫₁³ f(x)dx > 20

Answer

Explanation:

Step1: Recall Intermediate - Value Theorem

The Intermediate - Value Theorem states that if a function (y = f(x)) is continuous on a closed interval ([a,b]), and (k) is a number between (f(a)) and (f(b)), then there exists at least one number (c) in the interval ((a,b)) such that (f(c)=k). Here, (a = 1), (b = 3), (f(1)=10) and (f(3)=18).

Step2: Analyze option A

Just because (f(1) = 10) and (f(3)=18), we cannot say that (10\leq f(2)\leq18). The function is continuous, but it doesn't have to be monotonic. So, this statement is not necessarily true.

Step3: Analyze option B

We are only given the values of the function at the endpoints of the interval ([1,3]). We have no information about the derivative of the function, so we cannot conclude that the function is increasing on the interval ([1,3]).

Step4: Analyze option C

Since (10<17<18) and (f(x)) is continuous on ([1,3]), by the Intermediate - Value Theorem, there exists at least one (x) in the interval ((1,3)) (and thus in ([1,3])) such that (f(x)=17).

Step5: Analyze option D

Since (8<10), and (f(x)) is continuous on ([1,3]), there is no guarantee that (f(x)) will take the value (8) in the interval ((1,3)) as (8) is not between (f(1) = 10) and (f(3)=18).

Step6: Analyze option E

We know that (f(x)) is continuous on ([1,3]), but we don't have enough information to determine the value of the definite - integral (\int_{1}^{3}f(x)dx). For example, the function could be very close to (10) for most of the interval ([1,3]) and then jump to (18) near (x = 3), and (\int_{1}^{3}f(x)dx) may not be greater than (20).

Answer:

C. (f(x)=17) has at least one solution in the interval ([1,3])