Stationary Points - Part 2

(how the sign of the derivative changes near a maximum, minimum, or horizontal point of inflection)


We know, from the previous section that at a stationary point the derivative function equals zero, \(\frac{dy}{dx} = 0\). But on top of knowing how to find stationary points, it is important to know how to classify them, that is to know how to determine whether a stationary point is a maximum, a minimum, or a horizontal point of inflexion.

The Sign of the Derivative

What allows us to distinguish the different types of stationary points is the sign of the derivative, \(\frac{dy}{dx}\), on either side (left or right) of the stationary point.

This is explained in the following tutorial.

Maximum Points

As we move along a curve, from left to right, past a maximum point we'll always observe the following:

  • as we approach the maximum, from the left hand side, the curve is increasing (going higher and higher). Consequently the derivative is positive: \(\frac{dy}{dx}>0\).
  • when we reach the maximum point, the gradient equals to zero. This is a stationary point and therefore: \(\frac{dy}{dx}=0\).
  • as we leave the maximum, moving towards the right, the curve is decreasing. Consequently the derivative is negative: \(\frac{dy}{dx}<0\).
In summary we can state that a stationary point is a maximum point if the sign of the derivative varies as we saw above, which is summarized in the following table:

to the left of a maximum point at a maximum point to the right of a maximum point
\(\frac{dy}{dx}>0\) \(\frac{dy}{dx}=0\) \(\frac{dy}{dx}< 0\)

Minimum Points

As we move along a curve, from left to right, past a minimum point we'll always observe the following:

  • as we approach the minimum, from the left hand side, the curve is decreasing. Consequently the derivative is negative: \(\frac{dy}{dx}<0\).
  • when we reach the minimum point, the gradient equals to zero. This is a stationary point and therefore: \(\frac{dy}{dx}=0\).
  • as we leave the minimum, moving towards the right, the curve is increasing. Consequently the derivative is positive: \(\frac{dy}{dx}>0\).
In summary we can state that a stationary point is a minimum point if the sign of the derivative varies as we saw above, which is summarized in the following table:

to the left of a minimum point at a minimum point to the right of a minimum point
\(\frac{dy}{dx}<0\) \(\frac{dy}{dx}=0\) \(\frac{dy}{dx}> 0\)

Horizontal Points of Inflection

There are two types of horizontal points of inflection, increasing and decreasing. We discuss both here.

Increasing Horizontal Point of Inflection

As we move along a curve, from left to right, past an increasing horizontal point of inflection we'll always observe the following:

  • as we approach the increasing horizontal point of inflection, from the left hand side, the curve is increasing. Consequently the derivative is positive: \(\frac{dy}{dx}>0\).
  • when we reach the increasing horizontal point of inflection, the gradient equals to zero. This is a stationary point and therefore: \(\frac{dy}{dx}=0\).
  • as we leave the increasing horizontal point of inflection, moving towards the right, the curve is increasing. Consequently the derivative is positive again: \(\frac{dy}{dx}>0\).
In summary we can state that a stationary point is a increasing horizontal point of inflection if the sign of the derivative varies as we saw above, which is summarized in the following table:

to the left of a increasing horizontal point of inflection at a increasing horizontal point of inflection to the right of a increasing horizontal point of inflection
\(\frac{dy}{dx}>0\) \(\frac{dy}{dx}=0\) \(\frac{dy}{dx}> 0\)


Decreasing Horizontal Point of Inflection

As we move along a curve, from left to right, past an decreasing horizontal point of inflection we'll always observe the following:

  • as we approach the decreasing horizontal point of inflection, from the left hand side, the curve is decreasing. Consequently the derivative is negative: \(\frac{dy}{dx} < 0\).
  • when we reach the decreasing horizontal point of inflection, the gradient equals to zero. This is a stationary point and therefore: \(\frac{dy}{dx}=0\).
  • as we leave the decreasing horizontal point of inflection, moving towards the right, the curve is decreasing. Consequently the derivative is negative again: \(\frac{dy}{dx} < 0\).
In summary we can state that a stationary point is a decreasing horizontal point of inflection if the sign of the derivative varies as we saw above, which is summarized in the following table:

to the left of a decreasing horizontal point of inflection at a decreasing horizontal point of inflection to the right of a decreasing horizontal point of inflection
\(\frac{dy}{dx}<0\) \(\frac{dy}{dx}=0\) \(\frac{dy}{dx}< 0\)