To find the absolute maximum and minimum values of the function $f(x) = x^5 + 5x^4 - 35x^3$ on the interval $[-1, 4]$, we will follow these steps:

1. Find the critical points by taking the derivative and setting it to zero.
2. Evaluate the function at the critical points and at the endpoints of the interval.
3. Determine the absolute maximum and minimum values.

Step 1: Find the derivative and critical points

First, we compute the derivative of $f(x)$:

$f'(x) = 5x^4 + 20x^3 - 105x^2$

Next, we set the derivative equal to zero:

$5x^4 + 20x^3 - 105x^2 = 0$

Factoring out the common term:

$5x^2(x^2 + 4x - 21) = 0$

This gives us:

$5x^2 = 0 \quad \text{or} \quad x^2 + 4x - 21 = 0$

From $5x^2 = 0$, we find $x = 0$.

For the quadratic equation $x^2 + 4x - 21 = 0$, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-4 \pm \sqrt{16 + 84}}{2} = \frac{-4 \pm \sqrt{100}}{2} = \frac{-4 \pm 10}{2}$

Calculating the two roots:

1. $x = \frac{6}{2} = 3$
2. $x = \frac{-14}{2} = -7$ (not in the interval $[-1, 4]$)

Thus, the critical points within the interval $[-1, 4]$ are $x = 0$ and $x = 3$.

Step 2: Evaluate the function at critical points and endpoints

We now evaluate $f(x)$ at the critical points and at the endpoints $x = -1$ and $x = 4$.

• At $x = -1$: $f(-1) = (-1)^5 + 5(-1)^4 - 35(-1)^3 = -1 + 5 + 35 = 39$

• At $x = 0$: $f(0) = 0^5 + 5(0)^4 - 35(0)^3 = 0$

• At $x = 3$: $f(3) = 3^5 + 5(3)^4 - 35(3)^3 = 243 + 405 - 945 = -297$

• At $x = 4$: $f(4) = 4^5 + 5(4)^4 - 35(4)^3 = 1024 + 1280 - 5600 = -4296$

Step 3: Determine absolute maximum and minimum

Now we compare the values:

• $f(-1) = 39$
• $f(0) = 0$
• $f(3) = -297$
• $f(4) = -4296$

The absolute maximum value is $39$ at $x = -1$.

The absolute minimum value is $-4296$ at $x = 4$.

Summary

• Absolute maximum: $39$ at $x = -1$
• Absolute minimum: $-4296$ at $x = 4$

Do you want more details or have any questions? Here are some follow-up questions:

1. How does the behavior of the function change outside the interval $[-1, 4]$?
2. What is the second derivative of the function, and how can it be used to determine concavity?
3. Can you analyze the end behavior of the function as $x$ approaches infinity?
4. What are the implications of finding a maximum or minimum on a given interval?
5. How would you approach finding maxima and minima for more complex functions?

Tip: Always check the endpoints along with critical points when looking for absolute extrema on a closed interval!