Final answer:
To find the value of c that satisfies the mean value theorem for f(x) = cos(x) - sin(x), one must specify a continuous interval [a, b], find f'(x), and use the theorem's equation f'(c) = [f(b) - f(a)] / (b - a) to solve for c.
Step-by-step explanation:
To find all numbers c that satisfy the mean value theorem for f(x) = cos(x) - sin(x), we need to ensure that we are considering a closed interval [a, b] on which f is continuous and differentiable—requirements of the mean value theorem.
The mean value theorem states that there exists at least one number c in the open interval (a, b) such that f'(c) = [f(b) - f(a)] / (b - a). For f(x) = cos(x) - sin(x), the derivative f'(x) = -sin(x) - cos(x). So we look for a value c such that -sin(c) - cos(c) = [f(b) - f(a)] / (b - a).
To apply the mean value theorem accurately, one needs to specify the interval [a, b]. Without this, we cannot provide a specific number c. However, once the interval is given, we would calculate the value of f(b) - f(a) and then solve for c in the equation given above.
It is also important to note that in some cases, there may be more than one value of c that satisfies the theorem for a given interval.