**Answers:**

*when the first derivative is positive the function is…*increasing*when the function is decreasing the first derivative is…*negative*when the first derivative crosses the x axis the function has…*a horizontal tangent line and a change of behavior (from increasing to decreasing or from decreasing to increasing) , that is a local maximum or minimum. If the first derivative just touches the*x*-axis (without crossing it and so it’s tangent to the x-axis) then the function has an*inflection point*with a horizontal tangent line*.**when the function crosses the x axis the first derivative is…*any… (the y-value of the derivative is just the slope of the function’s tangent line in the crossing point)*when the function has a local maximum (or minimum) the first derivative…*crosses the*x*-axis (it’s*y*-value is 0, that is )*when the first derivative has a local maximum (or minimum) the second derivative…*crosses the x-axis (it’s y-value is 0, that is )*when the function is concave upwards (smiling) its second derivative is…*positivenegative*when the function is concave downwards (sad) its second derivative is…*the tangent line of the first derivative slope is zero and the function may have an*when the second derivative is zero the first derivative … and the function…**inflection point*.

If the second derivative crosses the*x*-axis then first derivative has a local maximum or minimum and there’s actually an*inflection point*for the function.

If the second derivative just touches the*x*-axis (without crossing it and so it’s tangent to the x-axis) then things are more complicated…the first derivative is increasing (decreasing) and the function has an upward (downward) concavity*when the second derivative is positive (negative) the first derivative …. and the function….*