Question

Determine the point at which the graph of the function below has a horizontal tangent line. (If the function has no horizontal tangent line, enter NONE.)
y =x3 + 3x

Answer 1

Slope of a horizontal line is zero. Derivative of the given function is 3x^2 + 3 which has no real roots. So there are NO horizontal tangents to the given function.

Answer 2

3x^2+3

Answer 3

right

Answer 4

Determine the point at which the graph of the function below has a horizontal tangent line. (If the function has no horizontal tangent line, enter NONE.)
y =x3 + 3x
(x, y)

Answer 5

Right the derivative is 3x^2 + 3 and when you equate it to zero (because the derivative means slope, and slope should be zero for a horizontal line), you get

3x^2 + 3 = 0
3x^2 = -3
x^2 = -1,
which gives complex (non real) roots.

Since the derivative can't equal zero for any value of x, that means there is no horizontal tangent to the given function for any real value of x.