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The problem asks you to determine the intervals where the function \( f(x) = -e^x (x - 2) \) is concave up or concave down and to identify any inflection points. To solve this, we need to follow these steps:

### Step 1: Find the First Derivative \( f'(x) \)
We need the first derivative to later find the second derivative.

Given:
\[ f(x) = -e^x (x - 2) \]

Use the **product rule** for differentiation:  
\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\]  
Here, \( f(x) = -e^x \) and \( g(x) = (x - 2) \).

Now:
- \( f'(x) = -e^x \) (derivative of \(-e^x\) with respect to \(x\))
- \( g'(x) = 1 \) (derivative of \(x - 2\))

Applying the product rule:
\[
f'(x) = -e^x (x - 2) + (-e^x) \cdot 1 = -e^x (x - 2 + 1) = -e^x (x - 1)
\]

### Step 2: Find the Second Derivative \( f''(x) \)
We differentiate the first derivative \( f'(x) = -e^x (x - 1) \) again, using the product rule.

- \( f'(x) = -e^x \) (derivative of \(-e^x\))
- \( g'(x) = 1 \) (derivative of \(x - 1\))

Now:
\[
f''(x) = -e^x (x - 1) + (-e^x)(1) = -e^x (x - 1 + 1) = -e^x \cdot x
\]

Thus:
\[
f''(x) = -x e^x
\]

### Step 3: Find Inflection Points
Inflection points occur where the second derivative changes sign. This happens when \( f''(x) = 0 \) or is undefined.

Set \( f''(x) = 0 \):
\[
-x e^x = 0
\]
Since \( e^x \) is never zero, the only solution is:
\[
x = 0
\]

### Step 4: Determine Concavity
We now check the sign of \( f''(x) = -x e^x \) on intervals around \( x = 0 \).

- For \( x < 0 \), say \( x = -1 \):
  \[ f''(-1) = -(-1) e^{-1} = \frac{1}{e} > 0 \]
  So, the function is **concave up** on \( (-\infty, 0) \).

- For \( x > 0 \), say \( x = 1 \):
  \[ f''(1) = -(1) e^{1} = -e < 0 \]
  So, the function is **concave down** on \( (0, \infty) \).

### Step 5: Conclusion
- **Concave up**: \( (-\infty, 0) \)
- **Concave down**: \( (0, \infty) \)
- **Inflection point**: \( x = 0 \) (where the concavity changes)

---

Do you have any questions or need further clarification?

### Related Questions:
1. How do you apply the product rule to more complicated functions?
2. What are some practical uses of inflection points?
3. Can a function have more than one inflection point?
4. How does \( e^x \) affect the shape of a graph?
5. What are other ways to test concavity besides the second derivative?

**Tip:** Always double-check the sign changes around inflection points carefully to avoid sign errors.

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. f(x) = -e^x (x - 2)

Learn how to find the intervals of concavity and the inflection points for the function f(x) = -e^x(x - 2). This calculus problem involves applying the product rule and analyzing the second derivative to determine concavity.

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