find all the values of x at which the graph of h(x) = x(e^(-3x)) has a horizontal tangent

To find all the values of \( x \) at which the graph of \( h(x) = x e^{-3x} \) has a horizontal tangent, we need to determine where the derivative of \( h(x) \) equals zero. A horizontal tangent occurs where the slope of the function (i.e., the derivative) is zero.

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

The function is a product of two functions, so we'll use the **product rule** to differentiate. The product rule states that for two functions \( f(x) \) and \( g(x) \):

\[
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
\]

In our case:

- \( f(x) = x \)
- \( g(x) = e^{-3x} \)

First, we differentiate each part:

- \( f'(x) = 1 \)
- \( g'(x) = \frac{d}{dx}(e^{-3x}) = -3e^{-3x} \)

Now, apply the product rule:

\[
h'(x) = f'(x)g(x) + f(x)g'(x)
\]
\[
h'(x) = (1)(e^{-3x}) + (x)(-3e^{-3x})
\]
\[
h'(x) = e^{-3x} - 3x e^{-3x}
\]

### Step 2: Set the derivative equal to zero

For a horizontal tangent, we need \( h'(x) = 0 \):

\[
e^{-3x} - 3x e^{-3x} = 0
\]

Factor out \( e^{-3x} \):

\[
e^{-3x}(1 - 3x) = 0
\]

### Step 3: Solve for \( x \)

Since \( e^{-3x} \) is never zero (the exponential function never equals zero), we only need to solve:

\[
1 - 3x = 0
\]
\[
3x = 1
\]
\[
x = \frac{1}{3}
\]

### Final Answer:
The graph of \( h(x) = x e^{-3x} \) has a horizontal tangent at \( x = \frac{1}{3} \).

Would you like more details on any step or have any other questions?

Here are 5 related questions to further your understanding:
1. How does the product rule work in differentiation?
2. What does a horizontal tangent tell us about the behavior of a function?
3. How can we interpret the significance of exponential decay in \( e^{-3x} \)?
4. What is the second derivative of \( h(x) = x e^{-3x} \)?
5. How would we find the points of inflection for this function?

**Tip:** When solving equations involving exponentials, keep in mind that exponential terms like \( e^{-3x} \) are always positive, simplifying the solving process!

Discover how to find the horizontal tangent of the function h(x) = x(e^(-3x)) by applying calculus concepts like the product rule and properties of exponential functions. This detailed guide walks you through finding where the derivative equals zero, a key skill in calculus. Ideal for students in Grades 11-12 or those taking Calculus I.

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