**Extrema and Curve Sketching
**

**
This material comes from Sections 3.1 and 3.2 of the textbook.
**

**The First Derivative**- If
*f*'(*x*)>0 on an interval then*f*(*x*) is rising on that interval. - If
*f*'(*x*)<0 on an interval then*f*(*x*) is falling on that interval. - If
*f*'(*c*)=0 then*f*(*x*) has a horizontal tangent line at the point (*c*,*f*(*c*)). - If
*c*is in the domain of*f*, and*f*'(*c*) equals zero or does not exist, then*c*is called first-order critical number. -
**First Derivative Test for Relative Extrema:**- If
*c*is a first-order critical number, and*f*'(*x*)>0 just to the left of*c*and*f*'(*x*)<0 just to the right of*c*, then the point (*c*,*f*(*c*)) is a relative maximum. - If
*c*is a first-order critical number, and*f*'(*x*)<0 just to the left of*c*and*f*'(*x*)>0 just to the right of*c*, then the point (*c*,*f*(*c*)) is a relative minimum.

- If

- If
**The Second Derivative**- If
*f*''(*x*)>0 on an interval then*f*(*x*) is concave up on that interval (and*f*'(*x*) is increasing). - If
*f*''(*x*)<0 on an interval then*f*(*x*) is concave down on that interval (and*f*'(*x*) is decreasing). - If
*c*is in the domain of*f*, and*f*''(*c*) equals zero or does not exist, then*c*is called a second-order critical number. - If
*c*is a second-order critical number and*f*'(*x*) changes sign from just before*c*to just after*c*, then the point (*c*,*f*(*c*)) is an inflection point (where*f*(*x*) changes concavity). -
**Second Derivative Test for Relative Extrema:**- If
*f*'(*c*)=0 and*f*''(*c*)>0 then the point (*c*,*f*(*c*)) is a relative minimum. - If
*f*'(*c*)=0 and*f*''(*c*)<0 then the point (*c*,*f*(*c*)) is a relative maximum. - If
*f*'(*c*)=0 and*f*''(*c*) equals zero or does not exist, then YOU DO NOT YET KNOW WHETHER THERE IS A RELATIVE EXTREMUM OR NOT AND YOU MUST USE SOME OTHER METHOD (SUCH AS THE FIRST DERIVATIVE TEST) TO FIND OUT.

- If

- If

**How to Use Calculus to Sketch the Graph of a Continuous Function
**

**
**

**Determine the domain of***f*; that is, the set of all*x*for which*f*(*x*) is defined.**Compute the derivative***f*'(*x*) and find the first-order critical numbers of*f*(where*f*'(*x*)=0 or*f*'(*x*) does not exist). Mark the first-order critical numbers on a number line restricted to reflect the domain of*f*. This partitions the domain of*f*(*x*) into a number of intervals. Determine the sign of*f*'(*x*) on each interval (e.g., by using test points).**Compute the second derivative***f*''(*x*) and find all second-order critical numbers (where*f*''(*x*)=0 or*f*''(*x*) does not exist). Mark the second-order critical numbers on the number line used in the previous step. Determine the sign of*f*''(*x*) on each interval of the domain determined by the second-order critical numbers (e.g., by using test points).**Determine the behavior of the graph of***f*in each of the intervals determined by the first- and second-order critical numbers.**For each first-order and second-order critical number***c*, find*f*(*c*) and plot the point (*c*,*f*(*c*)) on a coordinate plane. Plot any other key points (for instance, intercepts) that can be found easily.**Complete the sketch by using the information in the previous steps to join the plotted points with a smooth curve.**

Tue Oct 13 09:55:42 EDT 1998