Question

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.

Answer 1

If you plug x = 5/2 = 2.5 into f(x), you'll get 5.83 approximately as the output. If you plug in x = 4 into f(x), then you'll get 6.67 as your output. The question is this: "Is the value 6 in that output interval from 5.83 to 6.67?". The answer is "yes, the value of 6 is between those two mentioned values". So the intermediate value theorem does apply here. How do we find that value of c? We simply replace f(x) with 6 and isolate x. See the attached image for the steps on how to do that. The final result of that attached image below is x = 2 or x = 3, but I'm able to rule out x = 2 as its not in the interval [2.5, 4]. The only thing left is x = 3

Final Answer: 3