`y = x^3 + x` Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.

Textbook Question

Chapter 2, 2.2 - Problem 58 - Calculus of a Single Variable (10th Edition, Ron Larson).
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kalau | (Level 2) Adjunct Educator

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Take the derivative of `y=x^3+x` .

`y'= 3x^2+1`

For horizontal tangent lines, the slope is zero.  Set the derivative function to zero because we want to know where x is when the slope is zero.

`0=3x^2+1`

`-1 = 3x^2`

`x^2 = -1/3`

The parabola `x^2`  will never hit negative one third.  We can see this on the graph below.  This means we will not have a solution for x.

Therefore, there are NO POINTS where a horizontal tangent line exists.

 

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