To determine the absolute maximum and minimum values of the function $f(x) = x^7 e^{5x}$ on the interval $-3 \leq x \leq 4$, we need to follow these steps:

1. Compute the derivative of $f(x)$ to find the critical points within the interval.
2. Evaluate $f(x)$ at these critical points and at the endpoints of the interval.
3. Compare the values from step 2 to identify the absolute maximum and minimum.

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

For $f(x) = x^7 e^{5x}$, we use the product rule: $f'(x) = \frac{d}{dx}(x^7) \cdot e^{5x} + x^7 \cdot \frac{d}{dx}(e^{5x})$ $f'(x) = 7x^6 e^{5x} + x^7 \cdot 5e^{5x}$ $f'(x) = e^{5x} (7x^6 + 5x^7)$ Simplifying further: $f'(x) = e^{5x} x^6 (7 + 5x)$

Step 2: Find the critical points

The derivative $f'(x) = e^{5x} x^6 (7 + 5x) = 0$ when either:

• $e^{5x} = 0$ (which never occurs),
• $x^6 = 0$, giving $x = 0$,
• $7 + 5x = 0$, giving $x = -\frac{7}{5}$.

So, the critical points within the interval $-3 \leq x \leq 4$ are $x = 0$ and $x = -\frac{7}{5}$.

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

Now we evaluate $f(x)$ at $x = -3$, $x = 4$, $x = 0$, and $x = -\frac{7}{5}$.

1. At $x = -3$: $f(-3) = (-3)^7 e^{5(-3)} = -2187 e^{-15}$

2. At $x = 4$: $f(4) = 4^7 e^{5 \cdot 4} = 16384 e^{20}$

3. At $x = 0$: $f(0) = 0^7 e^{5 \cdot 0} = 0$

4. At $x = -\frac{7}{5}$: $f\left(-\frac{7}{5}\right) = \left(-\frac{7}{5}\right)^7 e^{5 \left(-\frac{7}{5}\right)} = -\frac{823543}{78125} e^{-7}$

Step 4: Compare values

• $f(-3) = -2187 e^{-15}$
• $f(4) = 16384 e^{20}$
• $f(0) = 0$
• $f\left(-\frac{7}{5}\right) = -\frac{823543}{78125} e^{-7}$

The absolute maximum value is $f(4) = 16384 e^{20}$, occurring at $x = 4$.

The absolute minimum value is $f(-3) = -2187 e^{-15}$, occurring at $x = -3$.

Would you like more detailed explanations on any part of this solution?

Related Questions:

1. How do we determine if a critical point is a maximum or minimum?
2. Why do we evaluate the function at the endpoints of the interval?
3. How does the exponential factor $e^{5x}$ affect the behavior of $f(x)$?
4. What is the significance of the derivative being zero at certain points?
5. How would the interval change affect the absolute maximum and minimum values?

Tip:

In optimization problems, always remember to check both critical points and endpoints in a closed interval to ensure you capture the absolute extremes.