This illustration is to help students realize 3 different relationships.
We can tell if f is increasing or decreasing by looking at the sign of the first derivative.
We can tell if f is concave up or concave down by looking at the sign of the second derivative.
We can tell if f is concave up or concave down by looking at whether the first derivative is increasing or decreasing.
Increasing and Decreasing are NOT the same thing as concave up and concave down. A function can be concave up and increasing or concave up and decreasing. Similarly a function can be concave down and increasing or concave down and decreasing.
