Where are the functions f1(x)=|sin(x)| and f2(x) = sin(|x|) differentiable? Use n as an arbitrary integer constant.)
f1(x) = |sin(x)| is differentiable for all x ≠______________
f2(x) = sin(|x|) is differentiable for all x ≠______________
To determine where the functions f1(x) = |sin(x)| and f2(x) = sin(|x|) are differentiable, we need to consider the differentiability of their respective components.
For f1(x) = |sin(x)|, we have the absolute value of sin(x). The function sin(x) is differentiable for all real numbers (x ∈ ℝ), so the absolute value of sin(x) will also be differentiable for all x in ℝ, except at the points where sin(x) = 0.
The function sin(x) = 0 when x is an integer multiple of π, i.e., x = nπ, where n is an integer. Therefore, f1(x) = |sin(x)| will not be differentiable at these points, so we have:
f1(x) = |sin(x)| is differentiable for all x ≠ nπ, where n is an integer.
Now, for f2(x) = sin(|x|), we have the sine of the absolute value of x. The function |x| is differentiable for all real numbers (x ∈ ℝ) except at x = 0 (the point where its derivative is not defined).
Therefore, f2(x) = sin(|x|) will also not be differentiable at x = 0 (since |x| is not differentiable there), so we have:
f2(x) = sin(|x|) is differentiable for all x ≠ 0.
In summary:
- f1(x) = |sin(x)| is differentiable for all x ≠ nπ, where n is an integer.
- f2(x) = sin(|x|) is differentiable for all x ≠ 0.