If speed decreases, the car is decelerating and the second derivative will be negative. In other words, if the second derivative is positive, the slope of the function is increasing and if the second derivative is negative, the slope of the function is decreasing. Consider the total profit curve shown below. After Q1, it appears that the slope of the curve is decreasing. This is defined as an inflection point.
Inflection points occur where the second derivative of the function is equal to zero.
When the second derivative is negative, the curve is turning downward and is said to be concave down. When the second derivative is positive, the curve is turning upward and is said to be concave up. It follows that local maximums occur when the second derivative is negative and local minimums occur when the second derivative is positve. In the graphic above, we see that the local maximum occurs at Q2 and that at Q2 the curve is turning downward, i.e. the second derivative is negative. Practice Problems Determine the first derivative and second derivative of each of the following equations in terms of x. 1.