Find all the values of x where the tangent line is horizontal.
f left parenthesis x right parenthesis equals 2 x cubed plus 36 x squared plus 192 x plus 5
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Part 1
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To find the values of \( x \) where the tangent line to the curve is horizontal, we need to find the points where the derivative of the function \( f(x) \) is equal to zero. The derivative of a function represents the slope of the tangent line at any point on the curve, and a horizontal tangent line has a slope of 0.

The given function is:

\[
f(x) = 2x^3 + 36x^2 + 192x + 5
\]

### Step 1: Differentiate the function
We will first differentiate \( f(x) \) to find \( f'(x) \):

\[
f'(x) = \frac{d}{dx} \left( 2x^3 + 36x^2 + 192x + 5 \right)
\]

Using the power rule for differentiation, we get:

\[
f'(x) = 6x^2 + 72x + 192
\]

### Step 2: Set the derivative equal to 0
To find the points where the tangent line is horizontal, set \( f'(x) = 0 \):

\[
6x^2 + 72x + 192 = 0
\]

### Step 3: Solve the quadratic equation
Divide the entire equation by 6 to simplify:

\[
x^2 + 12x + 32 = 0
\]

Now, we can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 12 \), and \( c = 32 \). Substituting these values into the formula:

\[
x = \frac{-12 \pm \sqrt{12^2 - 4(1)(32)}}{2(1)}
\]

\[
x = \frac{-12 \pm \sqrt{144 - 128}}{2}
\]

\[
x = \frac{-12 \pm \sqrt{16}}{2}
\]

\[
x = \frac{-12 \pm 4}{2}
\]

Thus, we get two solutions:

\[
x = \frac{-12 + 4}{2} = \frac{-8}{2} = -4
\]

\[
x = \frac{-12 - 4}{2} = \frac{-16}{2} = -8
\]

### Final Answer:
The values of \( x \) where the tangent line is horizontal are:

\[
x = -4 \quad \text{and} \quad x = -8
\]

Would you like further explanation or a detailed breakdown of any of the steps?

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**Related Questions:**
1. How do you find the slope of a tangent line at a given point on a curve?
2. What is the quadratic formula and how do you use it to solve equations?
3. How does the power rule of differentiation apply to polynomial functions?
4. How can you verify the correctness of a solution for a quadratic equation?
5. What is the relationship between a function's derivative and the shape of its graph?

**Tip:** Always simplify a quadratic equation before using the quadratic formula to make the calculations easier.

Find all the values of x where the tangent line is horizontal. f(x) = 2x^3 + 36x^2 + 192x + 5

This problem demonstrates how to find the values of x where the tangent line to the function f(x) = 2x^3 + 36x^2 + 192x + 5 is horizontal. Using differentiation and solving a quadratic equation, we find that x = -4 and x = -8. Perfect for advanced high school students (Grades 11-12).

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