How do you find the points where the graph of the function $y = (x^3) + x$ has horizontal tangents and what is the equation?

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Answer 1 · 2016-09-22T02:22:12

$y$ has no real points through which there are horizontal tangents

$y=x^3+x$

$y$ would have horizontal tangents at points where $y'(x) =0$ - if such points exist for $x in RR$

$y'(x) = 3x^2+1$

$y'(x) = 0 -> 3x^2+1=0$

$x=sqrt(-1/3) = +-1/sqrt3i$

Hence: $y'(x)=0$ has no $x in RR$

This can be seen by looking at the graph of $y$ below:

graph{x^3+x [-3.898, 3.896, -1.95, 1.948]}

How do you find the points where the graph of the function y = (x^3) + xy = (x^3) + x has horizontal tangents and what is the equation?