Curve Sketching and Derivatives

Consider a function; its first and second derivatives are used to sketch its curve.

The first derivative is zero when x equals four, and undefined when x equals zero or x equals six. These critical points define the intervals for analysis.

The sign of the first derivative determines if the curve is increasing or decreasing.

A sign change from negative to positive at x equals zero gives a local minimum.

A change from positive to negative at x equals four gives a local maximum. If the sign does not change at x equals six, no extremum exists there.

The second derivative is negative on the intervals from negative infinity to 0 and from 0 to 6, which shows that the plot is concave downward on these intervals.

But the second derivative is positive on the interval x greater than six, which gives the concave upward plot on this interval.

For example, consider a profit function. The first derivative tracks the profit's rise or fall and identifies its extreme values.

The second derivative shows how the rate of profit changes, which helps sketch the function.