The function's second derivative tells us many things about the math function itself. In this post, we'll explore its most essential meaning.
Let's start by recalling the first derivative. It is a measure of the rate of change of a function. On the other hand, the second derivative measures the rate of change of the first derivative (slope).
Concavity
The second derivative deals with the concavity of a function \(f\).
This characteristic has two possibilities: \(f\) is either concaving upward or downward. To evaluate what we mean by these two conditions, let's investigate the slope \(m\) of a curve:
- If the slope \(\frac{dy}{dx}\) keeps on increasing along an interval, the function concaves upward.
- If the slope \(\frac{dy}{dx}\) keeps on decreasing along an interval, the function concaves downward.
Since the second derivative measure the change in slope: \(y^{\prime \prime} = \Delta{y^{\prime}}\), we can interpret the result of the second derivative:
- If the change \(\Delta{y^{\prime}}\) is positive, or \(y^{\prime \prime}\) is greater than zero, then the slope between points increases; hence, the function concaves upward.
- If the change \(\Delta{y^{\prime}}\) is negative, or \(y^{\prime \prime}\) is less than zero, then the slope between points decreases; hence, the function concaves downward.
- If there is no change \(\Delta{y^{\prime}}\), or \(y^{\prime \prime}\) is zero, then the function is neither concaving upward nor downward.
