Final answer:
The absolute maximum value of the function f(x) = -x² + 4x + 4 on the interval [4,7] is 4, and the absolute minimum value is -17, evaluated at the endpoints since the critical point is not within the interval.
Step-by-step explanation:
To find the absolute maximum and minimum values of the function f(x) = -x² + 4x + 4 on the interval [4,7], we need to evaluate the function at critical points as well as at the endpoints of the interval. Since the function is a parabola opening downwards (because the coefficient of the x² term is negative), it has a single critical point which is a maximum. This is found by taking the derivative of f(x) and setting it to zero.
The derivative f'(x) = -2x + 4. Setting f'(x) = 0 we get x = 2. However, since 2 is not within our interval [4,7], we only need to evaluate f(x) at the endpoints of our interval.
At x = 4, f(4) = -4² + 4*4 + 4 = -16 + 16 + 4 = 4. At x = 7, f(7) = -7² + 4*7 + 4 = -49 + 28 + 4 = -17. Therefore, the absolute maximum value on the interval [4,7] is 4, and the absolute minimum value on the interval [4,7] is -17.