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.