Final answer:
The absolute maximum of the function f(x)=xe^(-x²) within the interval [0, infinity) occurs at x=1/√2, with the maximum value of (1/√2)e^(-1/2). The function has no absolute minimum on this interval.
Step-by-step explanation:
To find the absolute maxima and minima of the function f(x) = xe-x² on the interval [0, infinity), we must first determine its critical points by setting its derivative equal to zero. The derivative of f(x) is f'(x) = e-x² - 2x²e-x². Setting this equal to zero and solving for x gives us a critical point at x = 1/√2.
Next, we examine the value of f(x) at this critical point and at the endpoints of the interval. Since we are only considering the interval starting from zero to infinity, the only endpoint we need to consider is at x = 0. Evaluating the function at x = 0 and x = 1/√2, we get f(0) = 0 and f(1/√2) = (1/√2)e-(1/2), which is the maximum on this interval.
Lastly, we note that as x approaches infinity, xe-x² approaches zero, so there is no additional absolute maximum. There is also no absolute minimum on this interval as the function decreases towards zero as x increases, without reaching a point lower than f(0) = 0.