Final answer:
The absolute maximum value of the function f(x) = x/(x^2-4x+4) on the interval [-1,1] is at one of the endpoints since there are no critical points in the interval. After evaluation, the maximum value is found to be 1 at x=1, leading to the answer being option (b) 1.
Step-by-step explanation:
The function in question is f(x) = x/(x^2-4x+4). To find its absolute maximum value on the interval [-1,1], we need to consider the endpoints of the interval and any critical points within the interval. The denominator is a perfect square, (x-2)^2, which means our function can be simplified to be f(x) = x/(x-2)^2. Note that the denominator cannot be zero, so x=2 is not in our domain and within our interval.
First, let's evaluate the function at the endpoints of the interval:
- f(-1) = -1/((-1-2)^2) = -1/9
- f(1) = 1/((1-2)^2) = 1
Next, we must determine if there are any critical points in the interval by taking the derivative of f(x) and setting it equal to zero:
f'(x) = (x-2)^2 - 2x(x-2) / (x-2)^4 = 0
However, we can simplify noting that 2x-4 is the derivative of the denominator, ultimately showing our critical points will occur when the numerator is zero (since denominator cannot be zero in our domain).
Upon completing this simplification, we find no critical points in the interval [-1,1]. Therefore, the maximum value must occur at one of the endpoints. Comparing the values at -1 and 1, we can see that the absolute maximum value within the interval is 1. So, the answer to the question is (b) 1.