Find the absolute maximum of the function
f(x)=2x+1, where -2<x<0
(-x^2)+1, where x>0
To find the absolute maximum of the function, we first need to determine the critical points and the endpoints of the given interval.
Step 1: Find the critical points
The critical points are the values of x where the derivative of the function is either zero or undefined.
For the first function f(x) = 2x + 1, the derivative is constant (2) and never equals zero. Therefore, there are no critical points for this function.
For the second function f(x) = -x^2 + 1, the derivative is -2x. Set -2x equal to zero and solve for x:
-2x = 0
x = 0
So the critical point for the second function is x = 0.
Step 2: Find the endpoints
The given interval is -2 < x < 0. Thus, the endpoints are -2 and 0.
Step 3: Evaluate the function at the critical points and the endpoints
Now, we substitute the critical points and endpoints into the original function to find the corresponding y-values.
For the first function f(x) = 2x + 1:
At x = -2:
f(-2) = 2(-2) + 1 = -3
At x = 0:
f(0) = 2(0) + 1 = 1
For the second function f(x) = -x^2 + 1:
At x = 0 (the critical point):
f(0) = -(0^2) + 1 = 1
Step 4: Determine the absolute maximum
Comparing the y-values obtained, we can see that the maximum value is 1, which occurs at x = 0 for both functions. Therefore, the absolute maximum of the given function is 1.
In summary, the absolute maximum of the function f(x) = 2x + 1 for -2 < x < 0 and (-x^2) + 1 for x > 0 is 1, and it occurs at x = 0.