Question

show how to use the intermediate value theorem to show that the equation cos(\frac{\pi x}{3}) + 1 = \frac{7}{4} has a solution between 0 and 3.
let f(x) = cos(\frac{\pi x}{3}) + 1. in order for the intermediate value theorem to apply we must first check that f is continuous on the interval 0,3. you should verify this and be able to explain why it is the case.
we check the value of f at the left - endpoint of the interval 0,3:

Explanation:

Step1: Evaluate f(0)

Substitute \(x = 0\) into \(f(x)=\cos(\frac{\pi x}{3})+1\).
\[f(0)=\cos(0)+1\]
Since \(\cos(0) = 1\), then \(f(0)=1 + 1=2\).

Answer:

2