Find the absolute maximum value and the absolute minimum value, if any, of the function.
g(x) = -x(^2) + 2 x + 9
To find the absolute maximum and minimum values of a function, you need to find the critical points and the endpoints of the interval where the function is defined. To do that, you'll need to follow these steps:
Step 1: Find the derivative of the function g(x).
To find critical points, we need to find where the derivative of the function equals zero or is undefined. In this case, the derivative of g(x) = -x^2 + 2x + 9 is:
g'(x) = -2x + 2
Step 2: Set the derivative equal to zero and solve for x.
To find critical points, set g'(x) = 0 and solve for x:
-2x + 2 = 0
-2x = -2
x = 1
So, x = 1 is a critical point.
Step 3: Determine if there are any endpoints.
Check if the function is defined over a specific interval. If so, determine the values of g(x) at the endpoints of that interval. Since there is no information given about the interval, we'll assume it is over the set of real numbers.
Step 4: Evaluate the function at critical points and the endpoints.
Now substitute the critical points and the endpoints found in Step 2 and Step 3 into the original function g(x) to find the corresponding values of g(x):
g(1) = -(1^2) + 2(1) + 9
= -1 + 2 + 9
= 10
Step 5: Identify the absolute maximum and minimum values.
Compare the values of g(x) at the critical points and the endpoints to find the absolute maximum and minimum values. In this case, since there are no endpoints specified, the absolute maximum and minimum values will be at the critical point:
Absolute maximum value: 10
Absolute minimum value: 10