Question

Rolle Theorem f(x) = sin x + cos x. Find c ε [0,2,π]

Answer 1

To apply the Rolle's theorem to the function $f(x) = sin x + cos x$ in the interval $[0, 2\pi]$,

we need to check the conditions of the theorem: Continuity:

The function $f(x) = sin x + cos x$ is continuous on the interval $[0, 2\pi]$ as both sin x and cos x are continuous functions. Differentiability:

The function $f(x) = sin x + cos x$ is differentiable on the interval $(0, 2\pi)$ as both sin x and cos x are differentiable functions.

Function values: We have $f(0) = sin 0 + cos 0 = 0$ and $f(2π) = sin (2π) + cos (2π) = 0$.

Since all the conditions of Rolle's theorem are satisfied, there exists at least one point c in the interval $(0, 2\pi)$ such that $f'(c) = 0$.

To find the value of c, we need to find the derivative of f(x) and set it equal to zero: $f'(x) = cos x - sin x$ Setting $f'(x) = 0$,

we have $cos x - sin x = 0$. Simplifying further, $cos x = sin x$.

This equation is satisfied at $x = \frac{3π}{4}$.

Therefore, in the interval $[0, 2\pi]$, there exists at least one point $c = \frac{3π}{4}$ such that $f'(c) = 0$.