The first and second derivative tests.

1. Finding Maxima and Minima ...
The Second Derivative Test
HDR Lower Cave Carlsbad Caverns

2. Let ƒ(x) be a function which is continuous on the interval [-5, 5]. The
derivatives of ƒ(x) have the properties indicated in the table below. Draw a
sketch of a possible graph of ƒ(x). Assume ƒ(-1) = 0.

3. First Derivative Test. Suppose that c is a critical point of the function
ƒ and suppose that there is an interval (a, b) containing c.
• If ƒ '(x) > 0 for all x in (a, c) and ƒ '(x) < 0 for all x in (c, b), then c
is a local maximum of ƒ.
• If ƒ'(x) < 0 for all x in (a, c) and ƒ'(x) > 0 for all x in (c, b), then c is
a local minimum of ƒ.
First Derivative Test Practice
Can you use the Second Derivative
Test to answer these same questions?
If you need it here is some ...
Maxima and Minima Practice

4. The Second Derivative Test
Second Derivative Test. Suppose that c is a critical point of the
function ƒ and suppose that there is an interval (a, b) containing c.
• If ƒ'(c) = 0 and ƒ''(c) < 0 then c is a local maximum of ƒ.
• If ƒ'(c) = 0 and ƒ''(c) > 0 then c is a local minimum of ƒ.
First Derivative Test Practice

5. Use the second derivative test to find all local extrema.