document.write( "Question 472974: Find the maximum number of x-intercepts and the maximum number of turning points that the graph of the function can have.
\n" ); document.write( "f(x) = – x2 + x4 – x6 + 3 \n" ); document.write( "
Algebra.Com's Answer #324305 by robertb(5830)\"\" \"About 
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Applying Descartes' rule of signs, the following are the possibilities for the distribution of roots (3 variations of sign):
\n" ); document.write( "(i) 3 positive, 3 negative roots, 0 complex
\n" ); document.write( "(ii) 3 positive, 1 negative roots, 2 complex
\n" ); document.write( "(iii) 1 positive, 3 negative roots, 2 complex
\n" ); document.write( "(iv) 1 positive, 1 negative roots, 4 complex
\n" ); document.write( "Since the polynomial is symmetric, cases (ii) and (iii) are eliminated.\r
\n" ); document.write( "\n" ); document.write( "Taking the derivative, we get \"-6x%5E5+%2B+4x%5E3+-+2x+=+-2x%283x%5E4-2x%5E2+%2B+1%29\". Since the factor \"3x%5E4-2x%5E2+%2B+1\" does not have real roots, there is only 1 turning point, and it happens at x = 0. This means that there are 2 x-intercepts.\r
\n" ); document.write( "\n" ); document.write( "We verify graphically:\r
\n" ); document.write( "\n" ); document.write( "\"-x%5E6+%2B+x%5E4+-+x%5E2+%2B+3\"\r
\n" ); document.write( "\n" ); document.write( "\"graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+-x%5E6+%2B+x%5E4+-+x%5E2+%2B+3+%29\"
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