document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193571>Question 1193571</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the extrema of the ff. function on the given interval, if there are any. determine the values of x at which the extrema occur.<BR>\n" );
document.write( "f(x)= x the square root of 4-x^2 on [-1,2]. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848553 by </B><A HREF=/tutors/aboutme.mpl?userid=parmen><B>parmen(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=parmen><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=848553\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Sure, I've been improving my skills at solving these simplification problems. Let's find the extrema of the function:<BR>\n" );
document.write( "$$f(x)= x\sqrt{4-x^2}$$<BR>\n" );
document.write( "On the interval $[-1, 2]$.\r<BR>\n" );
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document.write( "We can find the extrema of a function by finding its critical points, which are the points where the derivative is zero or undefined.\r<BR>\n" );
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document.write( "**Steps to solve:**<BR>\n" );
document.write( "**1. Differentiate the function:**<BR>\n" );
document.write( "$$f'(x)=\sqrt{4-x^2}-\frac{x^2}{\sqrt{4-x^2}}$$\r<BR>\n" );
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document.write( "**2. Set the derivative equal to zero and solve for x:**<BR>\n" );
document.write( "$$f'(x)=0$$<BR>\n" );
document.write( "$$\sqrt{4-x^2}-\frac{x^2}{\sqrt{4-x^2}}=0$$<BR>\n" );
document.write( "$$\sqrt{4-x^2}=\frac{x^2}{\sqrt{4-x^2}}$$<BR>\n" );
document.write( "$$4-x^2=x^2$$<BR>\n" );
document.write( "$$2x^2=4$$<BR>\n" );
document.write( "$$x^2=2$$<BR>\n" );
document.write( "$$x=\pm\sqrt{2}$$\r<BR>\n" );
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document.write( "**3. Evaluate the function at the critical points and endpoints of the interval:**<BR>\n" );
document.write( "$$f(-1)=-1\sqrt{4-(-1)^2}=-\sqrt{3}$$<BR>\n" );
document.write( "$$f(-\sqrt{2})=-\sqrt{2}\sqrt{4-(-\sqrt{2})^2}=-2$$<BR>\n" );
document.write( "$$f(\sqrt{2})=\sqrt{2}\sqrt{4-(\sqrt{2})^2}=2$$<BR>\n" );
document.write( "$$f(2)=2\sqrt{4-2^2}=0$$\r<BR>\n" );
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document.write( "**4. Compare the values of the function at the critical points and endpoints to determine the extrema:**<BR>\n" );
document.write( "The maximum value of the function is $2$ at $x=\sqrt{2}$.<BR>\n" );
document.write( "The minimum value of the function is $-2$ at $x=-\sqrt{2}$.\r<BR>\n" );
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document.write( "**Answer:**<BR>\n" );
document.write( "The extrema of the function are:<BR>\n" );
document.write( "* Maximum value: $2$ at $x=\sqrt{2}$<BR>\n" );
document.write( "* Minimum value: $-2$ at $x=-\sqrt{2}$<BR>\n" );
document.write( " </SPAN>\n" );
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