How do you verify whether Rolle's theorem can be applied to the function f(x) = tan x in [0, π]?

To determine if Rolle's theorem applies to the function f(x) = tan x on the interval [0, π], we need to check three key conditions.

1. Continuity on the Closed Interval

Rolle's theorem requires that the function be continuous on the closed interval. The function tan x is continuous everywhere except where it is undefined. In the interval [0, π], tan x is undefined at x = π/2. Therefore, f(x) is not continuous on the entire interval.

2. Differentiability on the Open Interval

Next, we check if the function is differentiable on the open interval (0, π). Since tan x is differentiable wherever it is continuous, it is differentiable on (0, π) except at x = π/2.

3. Equal Values at the Endpoints

Finally, we need to see if the function takes the same value at the endpoints of the interval. We calculate:

• f(0) = tan(0) = 0
• f(π) = tan(π) = 0

Since f(0) = f(π), this condition is satisfied.

Final Assessment

Although the function meets the endpoint condition, it fails the continuity requirement due to the discontinuity at x = π/2. Therefore, Rolle's theorem cannot be applied to f(x) = tan x on the interval [0, π].