To find the absolute maximum value of the function $f(x) = x \ln(x)$ on the interval $x \in [e^{-2}, e]$, let's go through the following steps:

Step 1: Compute the Derivative of $f(x)$

The function is: $f(x) = x \ln(x)$ To find the critical points, we need to take the derivative of $f(x)$ with respect to $x$: $f'(x) = \ln(x) + 1$ Setting $f'(x) = 0$ to find critical points: $\ln(x) + 1 = 0$ $\ln(x) = -1$ $x = e^{-1}$

So, $x = e^{-1}$ is a critical point within the interval $[e^{-2}, e]$.

Step 2: Evaluate $f(x)$ at the Endpoints and Critical Point

Now, we need to evaluate $f(x)$ at $x = e^{-2}$, $x = e^{-1}$, and $x = e$:

1. At $x = e^{-2}$: $f(e^{-2}) = e^{-2} \ln(e^{-2}) = e^{-2} \cdot (-2) = -2e^{-2}$

2. At $x = e^{-1}$: $f(e^{-1}) = e^{-1} \ln(e^{-1}) = e^{-1} \cdot (-1) = -e^{-1}$

3. At $x = e$: $f(e) = e \ln(e) = e \cdot 1 = e$

Step 3: Determine the Maximum Value

The values of $f(x)$ at these points are:

• $f(e^{-2}) = -2e^{-2}$
• $f(e^{-1}) = -e^{-1}$
• $f(e) = e$

The maximum value among these is $f(e) = e$.

Conclusion

The absolute maximum value of $f(x) = x \ln(x)$ on the interval $[e^{-2}, e]$ is $\boxed{e}$, which corresponds to option (b).

Would you like a more detailed explanation of any step?

Here are some additional questions for further practice:

1. How do you find the derivative of a product of functions like $x \ln(x)$?
2. What is the significance of finding critical points when looking for extrema?
3. How do you determine if a critical point is a maximum or minimum?
4. Why do we evaluate the function at endpoints as well as critical points when finding absolute extrema?
5. How would the solution change if the interval was different, such as $[1, e]$?

Tip: Always evaluate a function at its endpoints as well as its critical points when finding absolute maximum or minimum values within a closed interval.