https://file.aiquickdraw.com/mathbot/file/17290546416178gz30hqa.jpg

Options are "local min" "local max" and "neither max or min"

We need to find the critical numbers for the function:

\[
f(x) = -4x^5 + 15x^4 + 20x^3 - 7
\]

Critical numbers occur where the first derivative of the function equals zero or where the derivative does not exist. Let's go step by step:

### Step 1: Find the first derivative of \( f(x) \)

To find the critical points, we first compute the derivative of \( f(x) \):

\[
f'(x) = \frac{d}{dx} \left( -4x^5 + 15x^4 + 20x^3 - 7 \right)
\]

Using the power rule for differentiation, we get:

\[
f'(x) = -20x^4 + 60x^3 + 60x^2
\]

### Step 2: Set \( f'(x) = 0 \)

Now, solve for \( x \) by setting the first derivative equal to zero:

\[
-20x^4 + 60x^3 + 60x^2 = 0
\]

Factor the equation:

\[
-20x^2(x^2 - 3x - 3) = 0
\]

This gives us two factors:

1. \( -20x^2 = 0 \)  → \( x = 0 \)

2. \( x^2 - 3x - 3 = 0 \)

### Step 3: Solve the quadratic equation

Solve the quadratic equation \( x^2 - 3x - 3 = 0 \) using the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-3)}}{2(1)}
\]

Simplifying:

\[
x = \frac{3 \pm \sqrt{9 + 12}}{2} = \frac{3 \pm \sqrt{21}}{2}
\]

So, the solutions are:

\[
x = \frac{3 + \sqrt{21}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{21}}{2}
\]

Numerically, these values are approximately:

\[
x \approx 3.791 \quad \text{and} \quad x \approx -0.791
\]

### Step 4: Classify the critical points using the second derivative test or graph

Now, the critical points are \( x = 0 \), \( x \approx 3.791 \), and \( x \approx -0.791 \). To classify them, either use the second derivative test or examine the behavior of the graph around these points. Based on the shape of the polynomial (as seen on a graph), we can classify them:

1. \( x = 0 \) is likely a **local minimum**.
2. \( x \approx -0.791 \) is likely a **local maximum**.
3. \( x \approx 3.791 \) is likely a **local minimum**.

### Final Answers:
- \( x \approx -0.791 \) is a local maximum.
- \( x = 0 \) is a local minimum.
- \( x \approx 3.791 \) is a local minimum.

Do you want further details, or have any questions?

Here are five related questions to deepen understanding:
1. How is the second derivative used to classify critical points?
2. What is the significance of factoring in finding critical numbers?
3. Can a function have more than one local maximum or minimum?
4. How does the power of \( x \) in a term influence the shape of a polynomial?
5. What are saddle points, and how do they relate to critical points?

**Tip:** When solving polynomials, always check the factors carefully; some critical points may result from higher-order terms.

Find the critical numbers of the function f(x) = -4x^5 + 15x^4 + 20x^3 - 7 and classify them by using the graph. Round your answers to three decimal places.

Learn how to find and classify the critical points of a polynomial function f(x) = -4x^5 + 15x^4 + 20x^3 - 7. This problem involves differentiation and finding the roots of the first derivative. Suitable for students in Grades 11-12.

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