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: f(0) = 2. and we check the value of f at the right endpoint of the interval 0,3: f(3) =

Explanation:

Step1: Recall the function

The function is $f(x)=\cos(\frac{\pi x}{3}) + 1$.

Step2: Evaluate at right - endpoint

Substitute $x = 3$ into $f(x)$. We know that $\cos(\frac{\pi\times3}{3})=\cos(\pi)=- 1$. Then $f(3)=\cos(\pi)+1=-1 + 1=0$.

Answer:

$0$