find the domain and all the critical points of the function.
a) x(the square root of
(4-x^2))
The domain is x must be less than or equal to positive or negative two but when i go to do the derivative to find the critical points i always get stuck. can someone show me step by step?
To find the domain and critical points of the function f(x) = x * √(4 - x^2), we need to consider two aspects.
1. Domain:
The function involves a square root, so the expression inside the square root (√(4 - x^2)) must be non-negative. We set this inequality:
4 - x^2 ≥ 0
To solve this inequality, we rearrange it as follows:
x^2 ≤ 4
Taking the square root of both sides (remembering to consider both positive and negative roots), we get:
x ≤ 2 and x ≥ -2
Therefore, the domain of the function is -2 ≤ x ≤ 2.
2. Critical Points:
Critical points occur where the derivative of the function is either zero or undefined. Let's find the derivative of f(x) using the product and chain rules:
f'(x) = √(4 - x^2) + x * (1/2) * (4 - x^2)^(-1/2) * (-2x)
Simplifying this expression, we get:
f'(x) = √(4 - x^2) - x^2 / √(4 - x^2)
Now, we set f'(x) equal to zero and solve for x:
√(4 - x^2) - x^2 / √(4 - x^2) = 0
Multiplying both sides by √(4 - x^2), we obtain:
4 - x^2 - x^2 = 0
Simplifying further, we find:
2x^2 = 4
Dividing by 2:
x^2 = 2
Taking the square root of both sides and considering both positive and negative roots, we have:
x = √2 and x = -√2
Therefore, the critical points of the function are x = √2 and x = -√2.
In summary, the domain of the function is -2 ≤ x ≤ 2, and the critical points are x = √2 and x = -√2.