If we look at the first derivative of a function, we can see that if the rate of change of the function is increasing on an interval, the function is concave up. Likewise, if the rate of change of the function is decreasing, the function is concave down.
Analytically Determining Concavity
We can determine whether the first derivative is increasing or decreasing by taking the derivative of the derivative (aka the second derivative). 🕵
Points of Inflection
If f"(a) = 0, then c is a special point called a point of inflection, which occurs when the function switches concavity. At a point of inflection, f(x) is neither concave up or concave down; it is linear at that point. 💡