Question

Used Consider the following function. f(x)=x−2sin(x) Find f′(x) f′(x)= For what values of x does the graph of f have a horizontal tangent? (Use n as your integer variable. Enter your answers as a comma-separated list.)

Answer 1

The derivative of f(x)=x−2sin(x) is f'(x) = 1 - 2cos(x). The graph of f has a horizontal tangent when cos(x) = 1/2, with solutions x = π/3 + 2πn and x = 5π/3 + 2πn, where n is an integer.

Step-by-step explanation:

To find the derivative of the function f(x)=x−2sin(x), we can use the sum and difference rule for derivatives. The derivative of x is 1 and the derivative of -2sin(x) is -2cos(x). Therefore, f'(x) = 1 - 2cos(x).

To find the values of x where the graph of f has a horizontal tangent, we need to find the values where f'(x) = 0. Setting 1 - 2cos(x) = 0 and solving for x, we get cos(x) = 1/2. The solutions in the interval 0 ≤ x ≤ 20 are x = π/3 + 2πn and x = 5π/3 + 2πn, where n is an integer.