Homework (3.4) Sketch the graph of $$ f(x)=2 x \mathrm{e}^{x} $$

Question

UpStudy AI · Accepted Answer

To sketch the graph of the function $$ f(x) = 2x e^{x} $$, let's analyze its key features, including its domain, intercepts, behavior, critical points, and concavity. ### 1. **Domain** The function $$ f(x) = 2x e^{x} $$ is defined for all real numbers since the exponential function $$ e^{x} $$ is defined everywhere. **Domain:** $$ (-\infty, \infty) $$ ### 2. **Intercepts** - **Y-intercept:** Set $$ x = 0 $$. $$ f(0) = 2 \cdot 0 \cdot e^{0} = 0 $$ So, the y-intercept is at $$ (0, 0) $$. - **X-intercept:** Same as the y-intercept since $$ f(0) = 0 $$. ### 3. **Behavior as $$ x \to \pm\infty $$** - **As $$ x \to -\infty $$:** $$ \lim_{x \to -\infty} 2x e^{x} = 0^{-} $$ The function approaches 0 from the negative side because $$ e^{x} $$ approaches 0 faster than $$ 2x $$ approaches $$ -\infty $$. - **As $$ x \to \infty $$:** $$ \lim_{x \to \infty} 2x e^{x} = \infty $$ The function grows without bound since both $$ x $$ and $$ e^{x} $$ increase. ### 4. **First Derivative and Critical Points** To find critical points, compute the first derivative $$ f'(x) $$. $$ f'(x) = \frac{d}{dx} [2x e^{x}] = 2e^{x} + 2x e^{x} = 2e^{x}(1 + x) $$ Set $$ f'(x) = 0 $$ to find critical points: $$ 2e^{x}(1 + x) = 0 $$ Since $$ e^{x} $$ is never zero, set $$ 1 + x = 0 $$ to find: $$ x = -1 $$ **Critical Point:** $$ x = -1 $$ **Function Value at Critical Point:** $$ f(-1) = 2(-1) e^{-1} = -\frac{2}{e} \approx -0.7358 $$ ### 5. **Second Derivative and Concavity** Compute the second derivative $$ f''(x) $$ to determine concavity. $$ f''(x) = \frac{d}{dx} [2e^{x}(1 + x)] = 2e^{x}(1 + x) + 2e^{x} = 2e^{x}(2 + x) $$ - **At $$ x = -1 $$:** $$ f''(-1) = 2e^{-1}(2 - 1) = \frac{2}{e} > 0 $$ Since the second derivative is positive, the function has a **local minimum** at $$ x = -1 $$. - **Concavity Intervals:** - **Concave Up:** When $$ f''(x) > 0 $$, i.e., $$ 2 + x > 0 $$ or $$ x > -2 $$. - **Concave Down:** When $$ f''(x) < 0 $$, i.e., $$ x < -2 $$. ### 6. **Summary of Key Features** - **Intercept:** Origin $$ (0, 0) $$ - **Local Minimum:** At $$ x = -1 $$, $$ f(-1) = -\frac{2}{e} $$ - **Asymptotic Behavior:** - Approaches 0 from below as $$ x \to -\infty $$ - Increases to $$ +\infty $$ as $$ x \to \infty $$ - **Increasing/Decreasing Intervals:** - **Increasing:** $$ x > -1 $$ - **Decreasing:** $$ x < -1 $$ - **Concavity:** - **Concave Up:** $$ x > -2 $$ - **Concave Down:** $$ x < -2 $$ ### 7. **Graph Sketch** While I can't provide a visual sketch here, you can imagine or plot the graph based on the following description: - The graph passes through the origin $$ (0, 0) $$. - It has a **local minimum** at $$ (-1, -\frac{2}{e}) $$, approximately $$ (-1, -0.7358) $$. - As $$ x $$ decreases beyond $$ -2 $$, the graph becomes concave down, approaching the x-axis from below. - Between $$ -2 $$ and $$ -1 $$, and for all $$ x > -1 $$, the graph is concave up. - For $$ x < -1 $$, the function is decreasing, reaching its minimum at $$ x = -1 $$, and then begins to increase for $$ x > -1 $$, rising sharply as $$ x $$ becomes large due to the exponential growth. ### 8. **Plotting Points for Reference** To aid in sketching, here are a few additional points: | $$ x $$ | $$ f(x) = 2x e^{x} $$ | |---------|-----------------------| | -2 | $$ 2(-2)e^{-2} = -\frac{4}{e^{2}} \approx -0.5413 $$ | | -1 | $$ -\frac{2}{e} \approx -0.7358 $$ | | -0.5 | $$ 2(-0.5)e^{-0.5} = -e^{-0.5} \approx -0.6065 $$ | | 0 | 0 | | 1 | $$ 2(1)e^{1} = 2e \approx 5.4366 $$ | | 2 | $$ 2(2)e^{2} = 4e^{2} \approx 29.556 $$ | ### 9. **Final Remarks** The function $$ f(x) = 2x e^{x} $$ combines linear and exponential growth, resulting in a graph that has a single local minimum and increases rapidly for positive $$ x $$. The interplay between the polynomial term $$ 2x $$ and the exponential term $$ e^{x} $$ creates interesting behavior, especially around $$ x = -1 $$ and $$ x = -2 $$. For an accurate graph, consider using graphing software or a graphing calculator by plotting the key points and understanding the shape based on the analysis above. To sketch the graph of $$ f(x) = 2x e^{x} $$, plot the function with the following key features: - **Intercept:** Origin $$ (0, 0) $$ - **Local Minimum:** At $$ x = -1 $$, $$ f(-1) = -\frac{2}{e} \approx -0.7358 $$ - **Asymptotic Behavior:** Approaches 0 from below as $$ x \to -\infty $$ and increases to $$ +\infty $$ as $$ x \to \infty $$ - **Concavity:** Concave down for $$ x < -2 $$ and concave up for $$ x > -2 $$ - **Increasing/Decreasing:** Decreasing for $$ x < -1 $$ and increasing for $$ x > -1 $$ Plotting these points and understanding the behavior will help in accurately sketching the graph.

Homework (3.4) Sketch the graph of \( f(x)=2 x \mathrm{e}^{x} \)

Real Tutor Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions