Finding Critical Numbers

    In general, the critical numbers divide the domain of a function into intervals on which the sign of the derivative remains the same, either positive or negative. Therefore, if a function is defined on that interval it is either increasing or decreasing on that interval. In particular the graph can not change directions on that interval. This is the crucial idea behind using the derivative to analyze graphs of function.

Definition (Critical Number) A critical number of a function is a real number in the domain of such that or does not exist.

Proposition (Critical Number Theorem) If is a continuous function and has a relative extremum at then is a critical number of

Example (Finding Critical Numbers) Find the critical numbers, if there are any, for the function



    Solution. The function is defined by
    


and a sketch of the graph is



Since, is not defined at the only critical number is   

Example (Finding Critical Numbers) Find the critical numbers, if there are any, for the function



    Solution. The function has derivative which is defined for all real numbers. Notice that at   and so 0 is a critical number. Also notice that is not a relative extrema nor an absolute extrema. In summary, critical numbers allow us to check if there are any extrema at that point, but not conversely.   

Example (Finding Critical Numbers) Find the critical numbers, if there are any, for the function

.

    Solution. The function has derivative which is defined for all real numbers. So to check for extrema we will need to determine where This occurs at where is any integer. In fact, in this case the absolute extrema of occurs at these values.  

Example (Finding Critical Numbers) Find the critical numbers, if there are any, for the function


    
    Solution. We have,



and so we need to check for any in the domain of such that or is undefined. We see that is undefined for and for So there are two critical numbers of and   

Example (Finding Critical Numbers) Find the critical numbers, if there are any, for the function



    Solution. We have,



and so we need to check for any in the domain of such that or is undefined. We see that is undefined for and for So there are three critical numbers of and   

Example (Finding Critical Numbers) Find the critical numbers, if there are any, for the function



    Solution. We have,



and so we need to check for any in the domain of such that or if is undefined. We see that is undefined for and for So there are three critical numbers of and   

Example (Finding Critical Numbers) Find the critical numbers, if there are any, for the function  


    
    Solution. We have,   and so we need to check for any in the domain of such that or is undefined. We see that is undefined for and there are no real numbers with One might be tempted to say that there is one critical numbers of However, is not a critical number because is not in the domain of

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Cite this as:
Finding Critical Numbers
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/finding-critical-numbers.html