https://file.aiquickdraw.com/mathbot/file/1726405942259ftqj4mcw.jpg

We are given the function:

\[
f(x) = x^3 - 1.5x^2 - 6x + 20
\]

We need to find the values of \( x \) where \( f'(x) = 0 \). Here's how we can proceed analytically:

### Step 1: Find \( f'(x) \)
Differentiate the function with respect to \( x \):

\[
f'(x) = \frac{d}{dx}\left(x^3 - 1.5x^2 - 6x + 20\right)
\]

Using the power rule for differentiation:
\[
f'(x) = 3x^2 - 3x - 6
\]

### Step 2: Set \( f'(x) = 0 \)
We now set the derivative equal to zero and solve for \( x \):

\[
3x^2 - 3x - 6 = 0
\]

Divide through by 3 to simplify the equation:

\[
x^2 - x - 2 = 0
\]

### Step 3: Solve the quadratic equation
Now solve the quadratic equation:

\[
x^2 - x - 2 = 0
\]

Factor the quadratic:

\[
(x - 2)(x + 1) = 0
\]

This gives the solutions:

\[
x = 2 \quad \text{or} \quad x = -1
\]

### Final Answer:
The values of \( x \) where \( f'(x) = 0 \) are \( x = -1 \) and \( x = 2 \), in increasing order.

Would you like any further details on the steps? 

Here are five related questions to expand on this:

1. How do you find the derivative of more complex functions like trigonometric or exponential ones?
2. What is the geometric significance of finding where \( f'(x) = 0 \)?
3. Can every cubic function have exactly two real critical points?
4. How does the power rule for differentiation generalize to polynomials of higher degree?
5. What if the quadratic equation couldn't be factored? How would you solve it then?

**Tip:** For quadratics that don't factor easily, the quadratic formula \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is a reliable method to find the roots.

If f(x) = x^3 - 1.5x^2 - 6x + 20, find analytically all values of x for which f'(x) = 0.

Discover how to solve for critical points by finding where f'(x) = 0 for the function x^3 - 1.5x^2 - 6x + 20. The solution involves differentiation and solving a quadratic equation. This detailed guide is ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation