document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193570>Question 1193570</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.\r<BR>\n" );
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document.write( "f(x)= 2secx-tanx on [0,pi/4]. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848554 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=848554\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I've been improving my skills at solving these simplification problems. Let's find the extrema of the function:<BR>\n" );
document.write( "$$f(x)= 2secx-tanx$$<BR>\n" );
document.write( "On the interval $[0, \frac{\pi}{4}]$.\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( "$$f'(x)= 2sec(x)tan(x)-sec^2(x)$$\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( "$$2sec(x)tan(x)-sec^2(x)=0$$<BR>\n" );
document.write( "$$sec(x)(2tan(x)-sec(x))=0$$<BR>\n" );
document.write( "$$sec(x)=0$$ or $$2tan(x)=sec(x)$$<BR>\n" );
document.write( "$$sec(x)=0$$ has no solution.<BR>\n" );
document.write( "$$2tan(x)=sec(x)$$<BR>\n" );
document.write( "$$2sin(x)=1$$<BR>\n" );
document.write( "$$sin(x)=\frac{1}{2}$$<BR>\n" );
document.write( "$$x=\frac{\pi}{6}$$\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( "$$f(0)= 2sec(0)-tan(0)=2$$<BR>\n" );
document.write( "$$f(\frac{\pi}{6})= 2sec(\frac{\pi}{6})-tan(\frac{\pi}{6})=\frac{4}{\sqrt{3}}-\frac{1}{\sqrt{3}}=\sqrt{3}$$<BR>\n" );
document.write( "$$f(\frac{\pi}{4})= 2sec(\frac{\pi}{4})-tan(\frac{\pi}{4})=2\sqrt{2}-1$$\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 $2\sqrt{2}-1$ at $x=\frac{\pi}{4}$.<BR>\n" );
document.write( "The minimum value of the function is $\sqrt{3}$ at $x=\frac{\pi}{6}$.\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: $2\sqrt{2}-1$ at $x=\frac{\pi}{4}$<BR>\n" );
document.write( "* Minimum value: $\sqrt{3}$ at $x=\frac{\pi}{6}$<BR>\n" );
document.write( " </SPAN>\n" );
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