Function derivatives (answers)

FunctionDerivatives 

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 y'\left( x \right)=0)
  • 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 y''\left( x \right)=0)
  • when the function is concave upwards (smiling) its second derivative is…
    positive
  • when the function is concave downwards (sad) its second derivative is…
    negative
  • when the second derivative is zero the first derivative … and the function…
    the tangent line of the first derivative slope is zero and the function may have an 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…
  • when the second derivative is positive (negative) the first derivative …. and the function….
    the first derivative is increasing (decreasing) and the function has an upward (downward) concavity

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