Determine the point(s) if any at which the graph of the function has a horizontal tangent line.
y=x^3+x
I graphed this function out and it looks like there isn't any horizontal tangents. Am I correct or am I forgeting something? I tried finding the derivitave of the function which ended up being 3x^2+1 but that equation will never equal to zero
You are correct. The slope is always positive.
see the graph at
http://www.wolframalpha.com/input/?i=x%5E3%2Bx
You're correct in finding the derivative of the function, which is 3x^2 + 1. To find the points where the graph of the function has a horizontal tangent line, we need to set the derivative equal to zero and solve for x. However, in this case, as you mentioned, the derivative 3x^2 + 1 will never equal zero because it is a quadratic function with no real solutions.
Since the derivative of the function is always positive, it means that the function is increasing for all values of x. Therefore, there are no points on the graph of the function where a horizontal tangent line exists.
So, your observation from the graph is correct – there are no horizontal tangents for the given function y = x^3 + x.