f(x) = 2xe^x. the graph is concave down when?

To find when the graph of \( f(x) = 2x e^x \) is concave down, we need to determine when the second derivative is negative.

### Step 1: Find the first derivative
First, we apply the product rule to differentiate \( f(x) = 2x e^x \).

\[
f'(x) = \frac{d}{dx} [2x e^x] = 2 \cdot e^x + 2x \cdot e^x = 2e^x (1 + x)
\]

### Step 2: Find the second derivative
Now we differentiate \( f'(x) = 2e^x(1 + x) \) again using the product rule.

\[
f''(x) = \frac{d}{dx} [2e^x (1 + x)] = 2e^x \cdot (1 + x) + 2e^x \cdot 1 = 2e^x (1 + x + 1) = 2e^x (x + 2)
\]

### Step 3: Set the second derivative less than zero
For concavity to be down, we need \( f''(x) < 0 \):

\[
2e^x (x + 2) < 0
\]

Since \( e^x \) is always positive, we only need to consider the factor \( (x + 2) \):

\[
x + 2 < 0
\]

\[
x < -2
\]

### Conclusion:
The graph of \( f(x) = 2x e^x \) is concave down when \( x < -2 \).

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does the graph behave at \( x = -2 \) in terms of concavity?
2. What does concavity tell us about the function’s inflection points?
3. How would you find the local maxima and minima of \( f(x) \)?
4. What is the significance of the first derivative in relation to concavity?
5. How would you analyze the end behavior of \( f(x) \)?

**Tip:** When checking concavity, always remember that positive second derivatives indicate concave up, while negative ones indicate concave down!

Find out when the graph of f(x) = 2xe^x is concave down by using the second derivative. Learn how to apply the product rule and the second derivative test, and analyze concavity for x < -2. Suitable for students in Grades 11-12 and early college.

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