document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1040466>Question 1040466</A>: <I><SPAN STYLE=\"word-break: break-all;\"> LAST PROBLEM!!!!!!!!!!!!!!!!!!!\r<BR>\n" );
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document.write( "2. Find the following, rounded to two decimal places.\r<BR>\n" );
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document.write( "Y=X4^(x)\r<BR>\n" );
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document.write( "(a) the intervals on which the function is increasing or decreasing.\r<BR>\n" );
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document.write( "(b) the range of the function \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #655281 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=655281\">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( "y = x*4^x<BR>\n" );
document.write( "dy/dx = 4^x+x*ln(4)*4^x ... apply derivative; use product rule and log rule<BR>\n" );
document.write( "dy/dx = 4^x * [ 1+x*ln(4) ] ... factor\r<BR>\n" );
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document.write( "Set dy/dx equal to zero. Then solve for x\r<BR>\n" );
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document.write( "4^x * [ 1+x*ln(4) ] = 0<BR>\n" );
document.write( "4^x = 0 or 1+x*ln(4) = 0\r<BR>\n" );
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document.write( "4^x = 0 has no solutions\r<BR>\n" );
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document.write( "let's solve 1+x*ln(4) = 0\r<BR>\n" );
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document.write( "1+x*ln(4) = 0<BR>\n" );
document.write( "x*ln(4) = -1<BR>\n" );
document.write( "x = -1/ln(4)<BR>\n" );
document.write( "x = -0.72134752044449\r<BR>\n" );
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document.write( "So dy/dx is only equal to zero when x = -0.72134752044449\r<BR>\n" );
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document.write( "If x is some smaller value, say x = -1, then \r<BR>\n" );
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document.write( "dy/dx = 4^x * [ 1+x*ln(4) ] <BR>\n" );
document.write( "dy/dx = 4^(-1) * [ 1+(-1)*ln(4) ] <BR>\n" );
document.write( "dy/dx =  -0.09657359027998\r<BR>\n" );
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document.write( "So dy/dx is negative for x values such that x < -0.72134752044449\r<BR>\n" );
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document.write( "This means the original function is decreasing when x < -0.72134752044449. So on the interval (-infinity, -0.72134752044449)<BR>\n" );
document.write( "Rounding to two decimal places gives <font color=red>(-infinity, -0.72)</font>\r<BR>\n" );
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document.write( "Let's plug in a value to the right of -0.72134752044449, say x = 0\r<BR>\n" );
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document.write( "dy/dx = 4^x * [ 1+x*ln(4) ] <BR>\n" );
document.write( "dy/dx = 4^(0) * [ 1+(0)*ln(4) ] <BR>\n" );
document.write( "dy/dx =  1\r<BR>\n" );
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document.write( "telling us that f(x) is increasing on the interval (-0.72134752044449, infinity)<BR>\n" );
document.write( "Rounding to two decimal places gives <font color=green>(-0.72, infinity)</font>\r<BR>\n" );
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document.write( "To wrap up part a) we found that...<BR>\n" );
document.write( "f(x) is decreasing on the interval <font color=red>(-infinity, -0.72)</font><BR>\n" );
document.write( "f(x) is increasing on the interval <font color=green>(-0.72, infinity)</font><BR>\n" );
document.write( "I'm using interval notation\r<BR>\n" );
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document.write( "--------------------------------------------------------------------\r<BR>\n" );
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document.write( "Part B)\r<BR>\n" );
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document.write( "In part A),we found that the function is decreasing on the interval from -infinity to -0.72134752044449. Then the function increases back again on the interval from -0.72134752044449 to infinity.\r<BR>\n" );
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document.write( "The graph will head downhill until it reaches x = -0.72134752044449. Then it will head back uphill. The question is: what is the corresponding y value for x = -0.72134752044449 ?\r<BR>\n" );
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document.write( "Let's find out by plugging in x = -0.72134752044449\r<BR>\n" );
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document.write( "y = x*4^x<BR>\n" );
document.write( "y = -0.72134752044449*4^(-0.72134752044449)<BR>\n" );
document.write( "y = -0.26536892271152<BR>\n" );
document.write( "y = <font color=blue>-0.27</font> ... rounding to two decimal places\r<BR>\n" );
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document.write( "The smallest y can get is -0.27\r<BR>\n" );
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document.write( "Therefore, the range in interval notation is <font color=blue>[-0.27, infinity)</font>\r<BR>\n" );
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document.write( "Take note how I used a square bracket to include the endpoint. \r<BR>\n" );
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document.write( "Here is a graph\r<BR>\n" );
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