Find the location of values of any global extrema of f(x)=1-x^2 on the interval {-2,2].
To find the location of values of any global extrema of the function f(x) = 1 - x^2 on the interval [-2,2], you can follow these steps:
1. Find the critical points:
The critical points are the values of x where the derivative of the function is either zero or undefined. To find the critical points, you need to find the derivative of f(x) with respect to x.
f'(x) = -2x
Set f'(x) equal to zero and solve for x:
-2x = 0
x = 0
2. Check the endpoints:
Since the interval is [-2,2], you need to check the function values at both -2 and 2. Substitute these values into the original function:
f(-2) = 1 - (-2)^2 = 1 - 4 = -3
f(2) = 1 - 2^2 = 1 - 4 = -3
-2 is not a critical point but is an endpoint of the interval. Note that the function is defined between -2 and 2, inclusive.
3. Evaluate the function at the critical points and endpoints:
So far, we have x = 0 as the only critical point. Evaluate the function at x = 0:
f(0) = 1 - 0^2 = 1 - 0 = 1
Therefore, the function has a global maximum at x = 0 with a value of 1, and a global minimum at both endpoints -2 and 2, with a value of -3.
To summarize, the global maximum is located at x = 0 with a value of 1, and the global minimum is located at both endpoints -2 and 2, with a value of -3.