Find the Horizontal Tangent Line e^x

Problem

ƒ(x)=ex

Solution

1. Identify the condition for a horizontal tangent line, which occurs when the derivative of the function is equal to zero.

ƒ(x)′=0

2. Differentiate the function ƒ(x)=ex with respect to x

d(ex)/d(x)=ex

3. Set the derivative equal to zero to solve for x

ex=0

4. Analyze the equation ex=0 Since the exponential function ex is always strictly greater than zero for all real values of x there are no solutions.

ex>0

5. Conclude that because the derivative is never zero, the function has no horizontal tangent lines.

Final Answer

The function *ƒ(x)=ex* has no horizontal tangent lines.

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