Final answer:
The function f(x) = cos x − sin x has its minima at x = 3π/4 and maxima at x = 7π/4, which are determined by setting the first derivative to zero and evaluating the second derivative at those points.
Step-by-step explanation:
The question is asking to find the maxima and minima of the function f(x) = cos x − sin x. To find these, we need to compute the derivative of the function and set it equal to zero to find the critical points.
Derivative of f(x) is f'(x) = -sin x - cos x. Setting the derivative equal to zero gives us f'(x) = 0, which simplifies to tan x = -1. This equation is true for x = 3π/4 and x = 7π/4, within one cycle of the trigonometric functions.
Now, we need to determine whether these points are maxima or minima by using the second derivative test or by examining the change of signs of the first derivative.
The second derivative f''(x) = -cos x + sin x evaluated at x = 3π/4 is positive, indicating a minimum, and evaluated at x = 7π/4 is negative, indicating a maximum. Therefore, the minima occur at x = 3π/4 and the maxima occur at x = 7π/4.