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 in the given intervals.
1. Evaluate at the critical point x = 3:
f(3) = -3(3) + 16 = 7
2. Evaluate at the endpoints:
f(1) = (1-2)^2 = 1
f(7) = -3(7) + 16 = -5
Comparing the values:
f(1) = 1
f(3) = 7
f(7) = -5
The absolute maximum value of the function is 7 at x = 3.