# What are the absolute extrema of #f(x)=x - e^x in[1,ln8]#?

##### 1 Answer

#### 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 *not* on the given interval

**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