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 \(f'(c) = 0\).