document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208061>Question 1208061</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given G(x) = -x^4 + 32x + 144,\r<BR>\n" );
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document.write( "A. Determine if G is even, odd,or even.\r<BR>\n" );
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document.write( "I think I have to evaluate G(-x). Yes? \r<BR>\n" );
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document.write( "B.There is a local maximum value of 400 at x = 4. <BR>\n" );
document.write( "Determine a second maximum value using a graph.\r<BR>\n" );
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document.write( "Note: DO NOT USE CALCULUS.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846218 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3838)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=846218\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "You can use the method tutor greenestamps mentioned. <ul><li>If all exponents of a polynomial function are even, then the function is even.</li><li>If all exponents of a polynomial function are odd, then the function is odd.</li><li>If the polynomial function has a mix of even and odd exponents, then the function is neither even nor odd.</li></ul>Your polynomial function is<BR>\n" );
document.write( "g(x) = -x^4 + 32x + 144<BR>\n" );
document.write( "which is the same as writing<BR>\n" );
document.write( "g(x) = -x^4 + 32x^1 + 144x^0<BR>\n" );
document.write( "The exponents are 4, 1, and 0. We have a mix of even and odd exponents, so <font color=red>the function is neither even nor odd</font>\r<BR>\n" );
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document.write( "Another approach for part (a)\r<BR>\n" );
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document.write( "Recall that:<ul><li>If g(x) = g(-x) for all x in the domain, then g(x) is even.</li><li>If g(-x) = -g(x) for all x in the domain, then g(x) is odd.</li></ul>The key thing to focus on here is the \"for all x\".<BR>\n" );
document.write( "If we can find one counterexample, then we'll disprove the claim.\r<BR>\n" );
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document.write( "Use a calculator, or do so by hand, to determine that<BR>\n" );
document.write( "g(1) = 175<BR>\n" );
document.write( "g(-1) = 111\r<BR>\n" );
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document.write( "g(1) = g(-1) is false so g(x) = g(-x) isn't true for all x in the domain.<BR>\n" );
document.write( "This means g(x) isn't an even function.\r<BR>\n" );
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document.write( "g(-1) = -g(1) being false means g(-x) = -g(x) is also false.<BR>\n" );
document.write( "This means g(x) isn't odd.\r<BR>\n" );
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document.write( "We conclude that <font color=red>g(x) is neither even nor odd</font>\r<BR>\n" );
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document.write( "If you wanted you can determine these facts<BR>\n" );
document.write( "g(x) = -x^4 + 32x + 144<BR>\n" );
document.write( "g(-x) = -(-x)^4 + 32(-x) + 144<BR>\n" );
document.write( "g(-x) = -x^4 - 32x + 144<BR>\n" );
document.write( "and,<BR>\n" );
document.write( "g(x) = -x^4 + 32x + 144<BR>\n" );
document.write( "-g(x) = -(-x^4 + 32x + 144)<BR>\n" );
document.write( "-g(x) = x^4 - 32x - 144<BR>\n" );
document.write( "Comparing g(x) with g(-x) shows the two functions aren't identical (since the +32x turned into -32x). Therefore, g(x) isn't even.<BR>\n" );
document.write( "Comparing g(-x) with -g(x) shows the two functions aren't identical. Therefore, g(x) isn't odd.<BR>\n" );
document.write( "A graphing calculator can be very useful here.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "There are many graphing options.<BR>\n" );
document.write( "Usually I would turn to <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a>, but <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> is actually the quicker choice when finding the local max.\r<BR>\n" );
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document.write( "Type the function into Desmos.<BR>\n" );
document.write( "Adjust the window so you can see the highest point. <BR>\n" );
document.write( "The window I'm using is<BR>\n" );
document.write( "xMin = -10<BR>\n" );
document.write( "xMax = 10<BR>\n" );
document.write( "yMin = -300<BR>\n" );
document.write( "yMax = 300<BR>\n" );
document.write( "Other window parameters are possible.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C11%2C-300%2C329%2C0%2C-x%5E4+%2B+32x+%2B+144%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-10%2C11%2C-300%2C329%2C0%2C-x%5E4+%2B+32x+%2B+144%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The curve is a weird lumpy hill of sorts. <BR>\n" );
document.write( "The curve isn't symmetric, but it somewhat resembles a parabola.\r<BR>\n" );
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document.write( "To find the highest point simply click anywhere on the curve. <BR>\n" );
document.write( "A small dot at the peak of this curve should show up. <BR>\n" );
document.write( "Click on that highest point to have the coordinates show up. <BR>\n" );
document.write( "You may have to click more than once.\r<BR>\n" );
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document.write( "<font color=red>The highest point on this curve is at (2,192)</font><BR>\n" );
document.write( "Therefore, g(x) reaches its max value of 192 when x = 2.\r<BR>\n" );
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document.write( "I'm not sure why your teacher mentioned \"There is a local maximum value of 400 at x = 4\" when that is false.<BR>\n" );
document.write( "It's possible your teacher made a typo somewhere?<BR>\n" );
document.write( "Note how g(4) = 16 and you can find larger outputs when trying something like g(0) and g(1).<BR>\n" );
document.write( "Also, this curve has one local max only. <BR>\n" );
document.write( "There aren't any neighborhoods we can zoom in on to get another local max.<BR>\n" );
document.write( "The region -infinity < x < 2 is always increasing while 2 < x < infinity is always decreasing.\r<BR>\n" );
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document.write( "If you want to use GeoGebra, then you'll need to use the function called \"max\". The template is<BR>\n" );
document.write( "max(f, a,b)<BR>\n" );
document.write( "where f is the function and a,b are the endpoints of the interval you are searching.<BR>\n" );
document.write( "One possible thing we could type in would be<BR>\n" );
document.write( "max( -x^4 + 32x + 144, 0, 4)<BR>\n" );
document.write( "The result GeoGebra produces is the point (1.9999999981, 192)<BR>\n" );
document.write( "We get a bit of rounding error. The x coordinate should be exactly 2. \r<BR>\n" );
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document.write( "If you want to use something like a TI83, then check out <a  rel=nofollow HREF=\"https://www.montgomerycollege.edu/_documents/academics/support/learning-centers/ackerman-learning-center-rockville/one-page-max-min.pdf\">this article</a><BR>\n" );
document.write( "Caution: The \"zoomFit\" feature might be tempting to use, but it's better to type in the xMin,xMax,yMin,yMax values by hand. \r<BR>\n" );
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document.write( "Want to do this without a graphing calculator? It would take a while, but you can make a table of values. <BR>\n" );
document.write( "It can be done entirely by hand (slowest method) or using a non-graphing calculator (slightly faster but still pretty slow).<BR>\n" );
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