To solve this problem systematically, we need to analyze the function $f(x) = e^{3x}(x+2)$ step by step.

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Step 1: Critical Numbers of $f(x)$

The critical numbers are found by setting $f'(x) = 0$ and solving for $x$.

Find $f'(x)$:

Using the product rule:

$f'(x) = \frac{d}{dx}[e^{3x}(x+2)] = e^{3x}(3)(x+2) + e^{3x}(1)$ $f'(x) = e^{3x} \left[ 3(x+2) + 1 \right]$ $f'(x) = e^{3x}(3x + 7)$

Solve $f'(x) = 0$:

Since $e^{3x} > 0$ for all $x$, we only solve $3x + 7 = 0$: $3x + 7 = 0 \implies x = -\frac{7}{3}$

Critical Numbers:

• $x = -\frac{7}{3}$ (where $f'(x) = 0$).
• $x = -2$ (where $f(x)$ is not defined because $x+2 = 0$).

Thus, $a = -\frac{7}{3}$ and $b = -2$.

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Step 2: Behavior Analysis

To analyze the behavior of $f(x)$ across the intervals, we study the signs of $f'(x)$ and $f''(x)$.

Intervals:

1. $x < a$ (i.e., $x < -\frac{7}{3}$).
2. $a < x < b$ (i.e., $-\frac{7}{3} < x < -2$).
3. $x > b$ (i.e., $x > -2$).

Concavity ($f''(x)$):

Find $f''(x)$ by differentiating $f'(x)$: $f'(x) = e^{3x}(3x + 7)$ $f''(x) = \frac{d}{dx}[e^{3x}(3x+7)] = e^{3x}(3)(3x+7) + e^{3x}(3)$ $f''(x) = e^{3x} \left[ 9x + 21 + 3 \right]$ $f''(x) = e^{3x}(9x + 24)$

The sign of $f''(x)$ depends on $9x + 24$: $9x + 24 = 0 \implies x = -\frac{8}{3}$

Thus, $f(x)$ changes concavity at $x = -\frac{8}{3}$.

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Step 3: Behavior and Shape Table

Using the above analysis, we complete the table:

$x$ Interval	Behavior/Shape
$-\infty$	Horizontal asymptote
$x < -\frac{7}{3}$	Decreasing, concave downward
$x = -\frac{7}{3}$	Local maximum
$-\frac{7}{3} < x < -2$	Increasing, concave downward
$x = -2$	Vertical asymptote
$x > -2$	Increasing, concave upward
$+\infty$	Horizontal asymptote

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Step 4: Key Points Summary

1. Critical numbers: $x = -\frac{7}{3}$ (local max), $x = -2$ (undefined, vertical asymptote).
2. Concavity change: $x = -\frac{8}{3}$.

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Let me know if you need clarifications or detailed graphs!
Related Questions:

1. How do you find the concavity of a function?
2. What is the significance of critical points?
3. Why is $e^{3x} > 0$ always true?
4. How do horizontal and vertical asymptotes differ?
5. Can a point of non-differentiability (like $x = -2$) also be a critical point?

Tip: Always check both the first and second derivatives to understand a function's shape fully!