Find all values of *x* (if any) where the tangent line to the graph of the given equation is horizontal. *HINT*: [The tangent line is horizontal when its slope is zero.] (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

*f*(*x*) = 9*x* − 9√x

To find all values of \( x \) where the tangent line to the graph of the function \( f(x) = 9x - 9\sqrt{x} \) is horizontal, we need to follow these steps:

1. **Find the derivative of the function \( f(x) \), since the derivative gives us the slope of the tangent line**. A horizontal tangent line occurs where the derivative equals 0.

\[
f(x) = 9x - 9\sqrt{x}
\]

2. **Differentiate** \( f(x) \) with respect to \( x \):

The derivative of \( 9x \) is simply 9, and for the second term, \( -9\sqrt{x} = -9x^{1/2} \), we use the power rule:

\[
\frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}
\]

Thus, the derivative is:

\[
f'(x) = 9 - 9 \cdot \frac{1}{2\sqrt{x}} = 9 - \frac{9}{2\sqrt{x}}
\]

3. **Set the derivative equal to zero**, since we want to find where the slope of the tangent line is zero:

\[
9 - \frac{9}{2\sqrt{x}} = 0
\]

4. **Solve for \( x \)**:

\[
9 = \frac{9}{2\sqrt{x}}
\]

Multiply both sides by \( 2\sqrt{x} \):

\[
18\sqrt{x} = 9
\]

Now, divide both sides by 18:

\[
\sqrt{x} = \frac{1}{2}
\]

Square both sides:

\[
x = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
\]

So, the value of \( x \) where the tangent line is horizontal is \( x = \frac{1}{4} \).

Thus, the answer is:

\[
x = \frac{1}{4}
\]

Would you like any further details or explanations?

### Related questions:
1. How do you find the slope of the tangent line for a given function?
2. Can there be multiple points where the tangent line is horizontal for some functions?
3. How do you differentiate functions involving radicals like \( \sqrt{x} \)?
4. What is the significance of the derivative in determining the shape of a graph?
5. How can you graphically interpret the result for \( f(x) = 9x - 9\sqrt{x} \)?

### Tip:
When solving equations involving square roots, always check for extraneous solutions, especially when dealing with restrictions on the domain of the function.

Find all values of x (if any) where the tangent line to the graph of the given equation is horizontal. HINT: [The tangent line is horizontal when its slope is zero.] (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x) = 9x − 9√x

Learn how to find the values of x where the tangent line to the graph of f(x) = 9x − 9√x is horizontal. This step-by-step guide covers key calculus concepts, including derivatives and the power rule, suitable for students in Grades 11-12 and early college.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation