Final answer:
The function f(x) = x + sin(2x) has infinitely many critical numbers. In the interval [-2,5], the critical numbers are x = π/6, x = 5π/6, x = π/6 + π, and x = 5π/6 + π.
Step-by-step explanation:
The function f(x) = x + sin(2x) has infinitely many critical numbers. Critical numbers occur when the derivative of the function is equal to zero or undefined. To find the critical numbers in the interval [-2,5], we need to find the values of x where f'(x) = 0 or f'(x) is undefined.
To find the derivative of f(x), we can use the sum rule and the chain rule. The derivative of x with respect to x is 1, and the derivative of sin(2x) with respect to x is 2cos(2x). So, the derivative of f(x) = x + sin(2x) is f'(x) = 1 + 2cos(2x).
Now, we need to solve the equation f'(x) = 0. Setting 1 + 2cos(2x) = 0 and solving for x gives us cos(2x) = -1/2. The values of x that satisfy this equation are x = π/6 + πn or x = 5π/6 + πn, where n is an integer.
Since we are only interested in the interval [-2,5], we need to find the values of x that fall within this interval. From the equation x = π/6 + πn, we can substitute n = 0, 1, 2, ... to find the values of x in the interval. Similarly, from the equation x = 5π/6 + πn, we can substitute n = 0, 1, 2, ... to find the values of x in the interval. The critical numbers in the interval [-2,5] are x = π/6, x = 5π/6, x = π/6 + π, and x = 5π/6 + π.