Find all x-values on [0, 2 $\pi$] where f(x) is not differentiable. f(x) = cos x cot x
B) 0, $\pi$, 2 $\pi$
D) 0
E) none

Question

Find all x-values on [0, 2 $\pi$] where f(x) is not differentiable. f(x) = cos x cot x
B) 0, $\pi$, 2 $\pi$ 
D) 0 
E) none

Quizplus · Accepted Answer

To find where f(x) is not differentiable, we need to find where the derivative of f(x) does not exist. First, we find the derivative of f(x) using the product rule: f'(x) = -sin(x)cot(x) - cos(x)csc^2(x). Note that f'(x) does not exist when cot(x) = 0 (because csc^2(x) = 1/sin^2(x), which is undefined at x = kπ for any integer k). This occurs when x = kπ + π/2 for any integer k.

Now we check the endpoints of the given interval [0, 2π]. Note that f(x) is defined and continuous on [0, 2π], so we only need to check the endpoints for differentiability. At x = 0, we have cot(0) = ∞, so f'(0) does not exist. At x = 2π, we have cot(2π) = ∞, so f'(2π) also does not exist. Therefore, the only x-values where f(x) is not differentiable are x = kπ + π/2 for any integer k, and the endpoints of the interval [0, 2π].

Plugging in the values for k, we get x = π/2, 3π/2, 5π/2, etc. The values for k that are within the interval [0, 2π] are k = 0 and k = 1, which give x = π/2 and x = 3π/2. Therefore, the only x-values in the interval [0, 2π] where f(x) is not differentiable are x = 0, π/2, 3π/2, and 2π. Converting π/2 and 3π/2 to decimal form, we get F1F1F1S1πF1F1F1 and F1F1F1S1πF1F1F10, respectively. Thus, the answer is B.

Find All X-Values on [0, 2 π\piπ ] Where F(x) Is Not Differentiable

Find All X-Values on [0, 2 $\pi$ ] Where F(x) Is Not Differentiable