Find the Horizontal Tangent Line y=x^3+9x

Problem

y=x3+9*x

Solution

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

2. Find the derivative of the function y=x3+9*x with respect to x using the power rule.

d(y)/d(x)=3*x2+9

3. Set the derivative equal to zero to solve for the xcoordinates where the slope is zero.

3*x2+9=0

4. Solve for x by isolating the x2 term.

3*x2=−9

x2=−3

5. Analyze the result to determine if any real solutions exist. Since the square of a real number cannot be negative (x2=−3, there are no real values of x that satisfy this equation.

6. Conclude that because there are no real solutions for x the function has no horizontal tangent lines.

Final Answer

No horizontal tangent lines

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