How to cal. The range of π+arcsin(3x-4x^3).give the whole solution step wise
To calculate the range of the given function π + arcsin(3x - 4x^3), we need to determine all possible values that this function can take. Here are the step-by-step instructions on how to find the range of the given function:
1. Start by considering the range of the arcsin function. The range of arcsin function is generally between -π/2 and π/2.
2. Let's first focus on the inner part of the function, which is (3x - 4x^3).
3. We need to determine the range of the inner function. To do this, we can find the minimum and maximum values of (3x - 4x^3), which will help us determine the range.
4. Take the derivative of the inner function, (3x - 4x^3), and set it equal to zero to find the critical points.
(d/dx)(3x - 4x^3) = 0
3 - 12x^2 = 0
12x^2 = 3
x^2 = 3/12
x^2 = 1/4
x = ±1/2
5. We have found two critical points: x = -1/2 and x = 1/2. We also need to consider the endpoints of the domain, which depend on the range of the arcsin function.
6. The range of the arcsin function is [-π/2, π/2]. Therefore, the values of (3x - 4x^3) should lie within this range to ensure that the arcsin function produces real values.
7. Substitute the critical points and endpoints into the inner function to check if they fall within the range of the arcsin function:
For x = -1/2, (3(-1/2) - 4(-1/2)^3) = (-3/2 + 1/2) = -1, which falls within the range [-π/2, π/2].
For x = 1/2, (3(1/2) - 4(1/2)^3) = (3/2 - 1/2) = 1, which also falls within the range [-π/2, π/2].
8. Now we can consider the outer function, which is π + arcsin(3x - 4x^3). Since the arcsin function takes values between -π/2 and π/2, the addition of π does not change the range of the function.
9. Hence, the range of the given function π + arcsin(3x - 4x^3) is [-π/2, π/2] shifted by the value π.
Therefore, the range of the given function is [π/2, 3π/2].