Find the absolute maximum, if it exists. f(x)= (x-2)^2 if 1≤x<3 and f(x)= -3x+16 if 3≤x<7
To find the absolute maximum, we need to evaluate the function at the critical points and endpoints.
1. Critical points:
The critical point occurs where the two pieces come together, so we need to find the value of x that makes (x-2)^2 = -3x+16:
(x-2)^2 = -3x+16
x^2 - 4x + 4 = -3x + 16
x^2 + x - 12 = 0
(x + 4)(x - 3) = 0
x = -4 or x = 3
Since x must be between 1 and 3, the critical point is x = 3.
2. Endpoints:
f(1) = (1-2)^2 = 1
f(3) = -3(3) + 16 = 7
Now we evaluate the function at the critical point and endpoints:
f(3) = 7
Therefore, the absolute maximum value of the function is 7.