Final answer:
To find the absolute minimum and maximum of f(x)=xe⁻²⁄₉₈ on [-5,14], one must calculate the derivative, find critical points, and evaluate f(x) at these points and the interval's endpoints. The function is likely to have a symmetry around the y-axis, and its graph should reflect critical points and decreasing behavior after reaching maximum value.
Step-by-step explanation:
The student is asking for the absolute minimum and maximum values of the function f(x)=xe⁻²⁄₉₈ on the interval [-5,14]. To find these values, one would typically calculate the derivative of f(x), set it equal to zero to find critical points, and evaluate the function at the critical points and endpoints of the interval. It’s important to note that not only do we look for points where the derivative equals zero, but also where it does not exist, because both can lead to potential maxima and minima.
Furthermore, since this function involves an exponent that includes x², the graph of f(x) would likely show a symmetrical pattern around the y-axis.
When graphing the function, we would label the graph with f(x) and x, scaling the axes appropriately to include the maximum and minimum values on both axes.
For example, if f(x)=10 when x=0, it’s essential to scale the y-axis to include this maximum value. The remark about the declining curve indicates that as x increases, f(x) is likely to decrease after reaching a certain maximal point.