Math Problem Statement
Solution
The given problem asks us to find the value of at which the graph of 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 is:
Step 2: Set the derivative equal to zero
To find where the tangent is horizontal, set the derivative equal to zero:
Step 3: Solve for
Solve the equation:
Take the natural logarithm (ln) of both sides:
Thus, the value of where the graph of has a horizontal tangent is .
Would you like more details on this problem, or do you have any questions?
Here are five related questions you might find interesting:
- What does the function look like graphically?
- How does the derivative of an exponential function like compare to other types of functions?
- How can you determine the concavity of the function at ?
- What is the geometric interpretation of a horizontal tangent line?
- How can you find the maximum or minimum values of the function ?
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.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Exponential Functions
Formulas
f'(x) = e^x - 2
Theorems
The point of horizontal tangency occurs where the derivative equals zero.
Suitable Grade Level
Grades 11-12
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