Extrema and Curve Sketching

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

1. 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.
2. 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.

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

1. Determine the domain of f; that is, the set of all x for which f(x) is defined.
2. 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).
3. 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).
4. Determine the behavior of the graph of f in each of the intervals determined by the first- and second-order critical numbers.

   

5. 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.
6. Complete the sketch by using the information in the previous steps to join the plotted points with a smooth curve.

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