We are asked to find the absolute extrema (i.e., the absolute maximum and minimum) of the function $f(x) = 2x^2 e^x$ on the closed interval $x \in [-3, 1]$. Let's proceed step by step.

Step 1: Find the derivative $f'(x)$

We apply the product rule to differentiate the function $f(x) = 2x^2 e^x$.

$f'(x) = \frac{d}{dx} \left( 2x^2 \cdot e^x \right)$ Using the product rule: $(uv)' = u'v + uv'$, where $u = 2x^2$ and $v = e^x$,

$f'(x) = 4x \cdot e^x + 2x^2 \cdot e^x$ We can factor out $2x e^x$:

$f'(x) = 2x e^x (2 + x)$

Step 2: Find the critical points

Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined. Since $f'(x) = 2x e^x (2 + x)$, we set this equal to zero:

$2x e^x (2 + x) = 0$

Since $e^x \neq 0$ for all real $x$, we solve:

$2x (2 + x) = 0$

This gives two solutions:

1. $x = 0$
2. $2 + x = 0 \quad \Rightarrow \quad x = -2$

So, the critical points are $x = 0$ and $x = -2$.

Step 3: Evaluate $f(x)$ at the critical points and endpoints

The absolute extrema must occur at either the critical points or the endpoints of the interval. The given interval is $x \in [-3, 1]$.

We now evaluate $f(x) = 2x^2 e^x$ at $x = -3, -2, 0, 1$.

• At $x = -3$: $f(-3) = 2(-3)^2 e^{-3} = 18 e^{-3} \approx 18 \times 0.0498 \approx 0.896$

• At $x = -2$: $f(-2) = 2(-2)^2 e^{-2} = 8 e^{-2} \approx 8 \times 0.1353 \approx 1.082$

• At $x = 0$: $f(0) = 2(0)^2 e^0 = 0$

• At $x = 1$: $f(1) = 2(1)^2 e^1 = 2e \approx 2 \times 2.718 \approx 5.436$

Step 4: Identify the absolute extrema

From the evaluations:

• $f(-3) \approx 0.896$
• $f(-2) \approx 1.082$
• $f(0) = 0$
• $f(1) \approx 5.436$

The absolute maximum is at $x = 1$, with value $f(1) \approx 5.436$.

The absolute minimum is at $x = 0$, with value $f(0) = 0$.

Final Answer

• Absolute maximum: $f(1) = 2e \approx 5.436$
• Absolute minimum: $f(0) = 0$

Would you like any further clarifications or detailed steps?

Related Questions:

1. How do you apply the product rule in general?
2. What happens to the derivative of $e^x$ for more complex functions?
3. Why do we only need to check endpoints and critical points on closed intervals?
4. How does the behavior of $e^x$ affect the growth rate of functions?
5. What are the differences between absolute and local extrema?

Tip: Always check the behavior of the function at endpoints and critical points when dealing with closed intervals.