FIRST & SECOND DERIVATIVE TEST
A. FIRST DERIVATIVE TEST
Let
1. Finding the critical points of f.
(Definition: A value is a critical point if
a. c is in the domain of the function , and
b. or does not exist)
2. Find the open interval(s) on which is increasing.
(Suppose that f is continuous at each point of , and differentiable at each point of . If for all , , then is increasing on )
3. Find the open interval(s) on which is decreasing
(Suppose that f is continuous at each point of , and differentiable at each point of . If for all , , then is decreasing on )
4. Find all relative maxima.
(Suppose f is defined on and c is a critical point, and if for x near and to the left of c, for x near and to the right of c, then is a relative maximum)
5. Find all relative minima
(Suppose f is defined on and c is a critical point, and if for x near and to the left of c, for x near and to the right of c, then is a relative minimum)
(NOTE: If the sign of is the same on both sides of c, then is not a relative extremum)
Exercise: Repeat the exercise above for
B. SECOND DERIVATIVE TEST
Let
1. Find where the graph of the function is concave up
(Suppose that f is twice-differentiable at each point of . If on , then the graph of f is concave up on )
2. Find where the graph of the function is concave down.
(Suppose that f is twice-differentiable at each point of . If on , then the graph of f is concave down on )
3. Find any points of inflection
(Definition: A point where the graph of a function has a tangent line and where the concavity changes is called a point of inflection)
(NOTE: At a point of inflection on the graph of a twice-differentiable function, . The converse is not true)
4. Use the second derivative tests to find the relative extrema.
( a. If and , then f has a relative maximum at ;
b. If and , then f has a relative minimum at )
(NOTE: The test fails if , if fails to exist, or if is hard to find)
Exercise: Repeat the exercise above for