Second+Derivative+Test

=__The Second Derivative Test__=

Like the First Derivative Test, the Second Derivative Test can be used to further predict the behavior of a curve.

//**Concavity and Points of Inflection**// The First Derivative Test was able to tell us the intervals on which f(x) was increasing or decreasing. The Second Derivative Test will be used to determine the intervals on which f(x) has positive and negative concavity.

If f(x) has positive concavity on an interval (a,b), then every point on (a,b) will lie above the graph of the tangent line to any point x within (a,b). In other words, "Concave Up, like a Cup" If f(x) has negative concavity on (a,b), then every point on (a,b) will lie Below the tangent line to every point x within (a,b). "Concave Down, like a Frown"

Points of Inflection are the points where f(x) switches concavity. The simplest algebraic method to find points of inflection is to set the second derivative, f ''(x), equal to 0, and solve for x. You must make certain to test surrounding points to ensure that the concavity is changing.

//**The Second Derivative Test**// We can determine the concavity using the Second Derivative Test

If f ''(a)>0, then f(x) is concave up at x=a. If f ''(a)<0, then f(x) is concave down at x=a.

This can be applied to determine if a critical point ( f '(x)=0 ), is a relative maximum or minimum. If f '(a)=0 and f ''(a)<0, then f(a) is a relative maximum. If f '(a)=0 and f ''(a)>0, then f(a) is a relative minimum.

Use the First and Second Derivative Tests to describe the behavior of the curve: f(x)=2x 2 +3x+1. Using the First Derivative Test, we find that f(x) is **Increasing on (-3/4)<x**, and is **Decreasing on x<(-3/4)**. (See Example Two from First Derivative Test Section) Using the power rule, we find the first derivative: f '(x)=4x+3. Using the power rule a second time, we find the second derivative: f ''(x)=4. As f ''(x) is positive for all values of x, we can say that **f(x) is concave up** for all values of x. As f(x) is concave up at the critical point x=(-3/4), we have shown that there is a **relative minimum at x=(-3/4)**.
 * Example One **