The sign of the second derivative shows information about the curve's shape:
if then is increasing
if then is decreasing
if then is stationary
1. Increasing Gradient
If then is increasing. This means that the gradient of the tangent kept on increasing and the curve becomes steeper, and the curve is concave upwards.
2. Decreasing Gradient
If then is decreasing. This means that the gradient of the tangent kept on decreasing and the curve becomes less steep, and the curve is concave downwards.
3. Stationary Points
If then is stationary. This means there is a change of the curve's concavity: from being concave upwards to downwards OR from being concave downwards to upwards. The curve goes through a changing concavity and has a point of inflexion.
Use the Geogebra app below - move point D to understand the concavity at that particular point. Click on the and checkbox to show the derivative and the second derivative. Have fun!
- Jesse Parete, 2012
Therefore, we can conclude that: