Question

Let f be the function given by f(x) = (9 * 2x * e^(-x/4)) / cos(x/2). The intermediate value theorem applied to f on the closed interval [24, 28] guarantees a solution in:

a. (0, π)
b. (1, 3)
c. (2, 4)
d. (3, 5)

Answer 1

The intermediate value theorem states that if a function is continuous on a closed interval [a, b], and the function takes on values f(a) and f(b) on that interval, then for any value between f(a) and f(b), there exists at least one value c in the interval [a, b] such that f(c) equals that value.

Step-by-step explanation:

The intermediate value theorem states that if a function is continuous on a closed interval [a, b], and the function takes on values f(a) and f(b) on that interval, then for any value between f(a) and f(b), there exists at least one value c in the interval [a, b] such that f(c) equals that value.

In this case, we have the function f(x) = (9 * 2x * e^(-x/4)) / cos(x/2), and we want to find a value between f(24) and f(28). To do this, we can evaluate f(24) and f(28) to find their values, and then determine the range of values between them using the intermediate value theorem.

Using a graphing calculator or software, we can find that f(24) is approximately 0.42 and f(28) is approximately 2.07. Therefore, there exists at least one value c in the interval [24, 28] such that f(c) is between 0.42 and 2.07. The answer is (a) (0, π).