Relation between original points and 1st/2nd derivative points

1. How does the second derivative relate to the original function?
2. What is the relationship between first derivative and second derivative?
3. What does the first and second derivative tell you?
4. What does the second derivative tell you about critical points?

How does the second derivative relate to the original function?

The units on the second derivative are “units of output per unit of input per unit of input.” They tell us how the value of the derivative function is changing in response to changes in the input. In other words, the second derivative tells us the rate of change of the rate of change of the original function.

What is the relationship between first derivative and second derivative?

Graphically the first derivative represents the slope of the function at a point, and the second derivative describes how the slope changes over the independent variable in the graph. For a function having a variable slope, the second derivative explains the curvature of the given graph.

What does the first and second derivative tell you?

Graphically, the first derivative gives the slope of the graph at a point. The second derivative tells whether the curve is concave up or concave down at that point. If the second derivative is positive at a point, the graph is bending upwards at that point.

What does the second derivative tell you about critical points?

The second derivative may be used to determine local extrema of a function under certain conditions. If a function has a critical point for which f′(x) = 0 and the second derivative is positive at this point, then f has a local minimum here.