What are the absolute extrema of #f(x)=x - e^x in[1,ln8]#?
1 Answer
There is an absolute maximum of
Explanation:
To determine absolute extrema on an interval, we must find the critical values of the function that lie within the interval. Then, we must test both the endpoints of the interval and the critical values. These are the spots where critical values could occur.
Finding critical values:
The critical values of
If:
Then:
So, the critical values will occur when:
Which implies that:
So:
The function's only critical value is at
Testing possible values:
Simply, find
#f(1)=1-e^1=1-eapprox-1.718#
#f(ln8)=ln8-e^ln8=ln8-8approx-5.921#
Thus, there is an absolute maximum of
Graphed is the original function on the given interval:
graph{x-e^x [.9, 2.079, -7, 1]}
Since there are no critical values, the function will remain decreasing over the entire interval. Since