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\n" ); document.write( "The requirements for f(x) are...
\n" ); document.write( "(1) has a local minimum
\n" ); document.write( "(2) has two negative intervals
\n" ); document.write( "(3) is an even function
\n" ); document.write( "From (3), we know the exponents on all variables must be even.
\n" ); document.write( "To get a local minimum, we want a rational function that is always positive between two vertical asymptotes. To get two vertical asymptotes in an even rational function, we can have a denominator of the form
\n" ); document.write( "
\n" ); document.write( "That function has vertical asymptotes at x=-a and x=a.
\n" ); document.write( "Given asymptotes at x=-a and x=a, to get a local minimum we want the function to be positive everywhere between x=-a and x=a; in particular, we want it to be positive at x=0. If we put just a positive constant in the numerator, then the function value is negative at x=0, so we can put a negative constant in the numerator.
\n" ); document.write( "So we want a function of the form
\n" ); document.write( "
\n" ); document.write( "where a and b are positive constants.
\n" ); document.write( "That function has the following characteristics:
\n" ); document.write( "(1) It is even
\n" ); document.write( "(2) It has vertical asymptotes at x=-a and x=a
\n" ); document.write( "(3) Its value is positive at x=0
\n" ); document.write( "(4) With vertical asymptotes at x=-a and x=a and a positive value at x=0, it has a local minimum
\n" ); document.write( "(5) When x is less than -a or greater than a, the function value is negative, so its graph has two negative intervals
\n" ); document.write( "So all the requirements are met.
\n" ); document.write( "After playing with the numbers to find a function with a nice graph, I came up with the function
\n" ); document.write( "
\n" ); document.write( "Here is a graph, showing all the required characteristics....
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