Find the Horizontal Tangent Line y=2+8x-x^2

Problem

y=2+8*x−x2

Solution

1. Identify the condition for a horizontal tangent line, which occurs when the slope of the function is zero, meaning the derivative d(y)/d(x)=0

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

d(y)/d(x)=d(2+8*x−x2)/d(x)

d(y)/d(x)=8−2*x

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

8−2*x=0

4. Solve for x by isolating the variable.

−2*x=−8

x=4

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

y=2+8*(4)−(4)2

y=2+32−16

y=18

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

Final Answer

y=18

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