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The given problem asks us to find the value of \( x \) at which the graph of \( f(x) = e^x - 2x \) has a horizontal tangent. To solve this, we need to find the derivative of the function and set it equal to zero, since a horizontal tangent occurs where the slope (the derivative) is zero.

### Step 1: Differentiate the function
The derivative of \( f(x) = e^x - 2x \) is:

\[
f'(x) = \frac{d}{dx}(e^x) - \frac{d}{dx}(2x) = e^x - 2
\]

### Step 2: Set the derivative equal to zero
To find where the tangent is horizontal, set the derivative equal to zero:

\[
e^x - 2 = 0
\]

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

\[
e^x = 2
\]

Take the natural logarithm (ln) of both sides:

\[
x = \ln(2)
\]

Thus, the value of \( x \) where the graph of \( f(x) \) has a horizontal tangent is \( x = \ln(2) \).

Would you like more details on this problem, or do you have any questions?

Here are five related questions you might find interesting:
1. What does the function \( f(x) = e^x - 2x \) look like graphically?
2. How does the derivative of an exponential function like \( e^x \) compare to other types of functions?
3. How can you determine the concavity of the function \( f(x) \) at \( x = \ln(2) \)?
4. What is the geometric interpretation of a horizontal tangent line?
5. How can you find the maximum or minimum values of the function \( f(x) = e^x - 2x \)?

**Tip:** When working with exponentials, it's helpful to recall that taking the natural logarithm helps solve equations where the variable is in the exponent.

For what value of x does the graph of f(x) = e^x - 2x have a horizontal tangent?

Discover how to find the value of x where the graph of f(x) = e^x - 2x has a horizontal tangent. This solution involves calculating the derivative and solving for x, a key concept in calculus. Ideal for students in Grades 11-12.

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