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

Problem

y=x3+3*x

Solution

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

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

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

3. Set the derivative equal to zero to find the xvalues where the slope is horizontal.

3*x2+3=0

4. Solve for x by isolating the variable.

3*x2=−3

x2=−1

5. Conclude that since x2=−1 has no real solutions, there are no points on the curve where the tangent line is horizontal.

Final Answer

No horizontal tangent lines exist for *y=x3+3*x

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