Answer :
Since we are only concerned with the function's behavior within the interval [-1, 4]. The absolute extrema will occur either at the critical points within this interval or at the endpoints themselves.
The absolute maximum and absolute minimum values of the function f(x) = x * e^(-x^2/8) on the interval [-1, 4] can be found by evaluating the function at its critical points and endpoints.
To find the critical points, we need to find where the derivative of the function is equal to zero or does not exist. Taking the derivative of f(x) with respect to x:
f'(x) = e^(-x^2/8) - (x^2/4) * e^(-x^2/8)
Setting f'(x) equal to zero and solving for x is a complex process involving numerical methods. Therefore, we can utilize a graphing calculator or software to find the critical points.
By evaluating the function f(x) at the critical points and endpoints of the interval [-1, 4], we can determine the absolute maximum and minimum values. Comparing the function values at these points, we can identify the highest and lowest values.
To find the absolute maximum and minimum values of a function on a closed interval, we need to consider the critical points and endpoints of the interval.
The critical points occur where the derivative of the function is equal to zero or does not exist. In this case, finding the derivative of f(x) is not straightforward due to the presence of the exponential function. Therefore, we can use numerical methods or graphing software to determine the critical points.
By evaluating the function f(x) at the critical points and the endpoints of the interval [-1, 4], we obtain a set of function values. Comparing these values allows us to identify the absolute maximum and minimum values.
For example, we can evaluate f(x) at x = -1, x = 4, and the critical points. The highest function value among these points represents the absolute maximum, while the lowest function value represents the absolute minimum.
It is worth noting that in some cases, the critical points may lie outside the given interval. However, since we are only concerned with the function's behavior within the interval [-1, 4], the absolute extrema will occur either at the critical points within this interval or at the endpoints themselves.
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