Find the Horizontal Tangent Line y=x^2+3x

Problem

y=x2+3*x

Solution

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

d(y)/d(x)=0

2. Differentiate the function with respect to x using the power rule.

d(y)/d(x)=2*x+3

3. Set the derivative equal to zero to find the xcoordinate of the point of tangency.

2*x+3=0

4. Solve for x by subtracting 3 and dividing by 2.

2*x=−3

x=−3/2

5. Substitute the xvalue back into the original equation to find the corresponding ycoordinate.

y=(−3/2)2+3*(−3/2)

y=9/4−9/2

y=−9/4

6. Determine the equation of the horizontal line, which is of the form y=k where k is the ycoordinate found.

y=−9/4

Final Answer

y=−9/4

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