document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209901>Question 1209901</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let<BR>\n" );
document.write( "f(x) = \frac{x^4 + 2x^3 + 3x^2 + 2x + 1}{x}.<BR>\n" );
document.write( "Find the minimum value of x for x > 0.<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 #851011 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851011\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's find the minimum value of $f(x)$ for $x > 0$.\r<BR>\n" );
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document.write( "**1. Simplify the Expression**\r<BR>\n" );
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document.write( "$$f(x) = \frac{x^4 + 2x^3 + 3x^2 + 2x + 1}{x} = x^3 + 2x^2 + 3x + 2 + \frac{1}{x}$$\r<BR>\n" );
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document.write( "**2. Apply AM-GM Inequality**\r<BR>\n" );
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document.write( "We want to find the minimum value of this expression for $x > 0$. We can use the AM-GM inequality, which states that for non-negative numbers $a_1, a_2, \dots, a_n$, we have:\r<BR>\n" );
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document.write( "$$\frac{a_1 + a_2 + \dots + a_n}{n} \ge \sqrt[n]{a_1 a_2 \dots a_n}$$\r<BR>\n" );
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document.write( "Let's rewrite the expression as:\r<BR>\n" );
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document.write( "$$f(x) = x^3 + 2x^2 + 3x + 2 + \frac{1}{x} = x^3 + x^2 + x^2 + x + x + x + 1 + 1/x$$\r<BR>\n" );
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document.write( "Applying AM-GM to the terms $x^3, x^2, x^2, x, x, x, 1, 1/x$, we have:\r<BR>\n" );
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document.write( "$$\frac{x^3 + x^2 + x^2 + x + x + x + 1 + 1/x}{8} \ge \sqrt[8]{x^3 \cdot x^2 \cdot x^2 \cdot x \cdot x \cdot x \cdot 1 \cdot \frac{1}{x}} = \sqrt[8]{x^8} = x$$\r<BR>\n" );
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document.write( "This inequality doesn't directly help us find the minimum value.\r<BR>\n" );
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document.write( "Let's try a different approach. We can rewrite the function as:\r<BR>\n" );
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document.write( "$$f(x) = x^3 + 2x^2 + 3x + 2 + \frac{1}{x}$$\r<BR>\n" );
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document.write( "Let's try to find a way to apply AM-GM effectively. Notice that $x^3 + \frac{1}{x}$ is not a good combination.\r<BR>\n" );
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document.write( "Instead, let's rewrite the expression as:\r<BR>\n" );
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document.write( "$$f(x) = x^3 + 2x^2 + 3x + 2 + \frac{1}{x}$$\r<BR>\n" );
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document.write( "Let's consider the expression:\r<BR>\n" );
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document.write( "$$f(x) = x^3 + x^2 + x^2 + x + x + x + 1 + \frac{1}{x}$$\r<BR>\n" );
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document.write( "This doesn't seem to lead to a simple solution.\r<BR>\n" );
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document.write( "Let's try a different approach.\r<BR>\n" );
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document.write( "We can rewrite $f(x)$ as:\r<BR>\n" );
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document.write( "$$f(x) = x^3 + 2x^2 + 3x + 2 + \frac{1}{x}$$\r<BR>\n" );
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document.write( "Let's find the derivative of $f(x)$:\r<BR>\n" );
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document.write( "$$f'(x) = 3x^2 + 4x + 3 - \frac{1}{x^2}$$\r<BR>\n" );
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document.write( "To find the minimum value, we set $f'(x) = 0$:\r<BR>\n" );
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document.write( "$$3x^2 + 4x + 3 = \frac{1}{x^2}$$<BR>\n" );
document.write( "$$3x^4 + 4x^3 + 3x^2 = 1$$<BR>\n" );
document.write( "$$3x^4 + 4x^3 + 3x^2 - 1 = 0$$\r<BR>\n" );
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document.write( "Let's see if we can find a root. Let's try $x = 0.5$.\r<BR>\n" );
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document.write( "$$3(0.5)^4 + 4(0.5)^3 + 3(0.5)^2 - 1 = 3(1/16) + 4(1/8) + 3(1/4) - 1 = 3/16 + 1/2 + 3/4 - 1 = (3 + 8 + 12 - 16)/16 = 7/16 > 0$$\r<BR>\n" );
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document.write( "Let's try $x = 0.4$.\r<BR>\n" );
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document.write( "$$3(0.4)^4 + 4(0.4)^3 + 3(0.4)^2 - 1 = 3(0.0256) + 4(0.064) + 3(0.16) - 1 = 0.0768 + 0.256 + 0.48 - 1 = -0.1872 < 0$$\r<BR>\n" );
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document.write( "Since $f'(0.4) < 0$ and $f'(0.5) > 0$, there is a root between 0.4 and 0.5.\r<BR>\n" );
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document.write( "Let's try $x = 0.45$.\r<BR>\n" );
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document.write( "$$3(0.45)^4 + 4(0.45)^3 + 3(0.45)^2 - 1 \approx 0.0410 + 0.3645 + 0.6075 - 1 = 0.013$$\r<BR>\n" );
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document.write( "So the root is close to 0.45.\r<BR>\n" );
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document.write( "Let's try $x = 0.44$.\r<BR>\n" );
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document.write( "$$3(0.44)^4 + 4(0.44)^3 + 3(0.44)^2 - 1 \approx 0.0354 + 0.3400 + 0.5808 - 1 = -0.0438$$\r<BR>\n" );
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document.write( "So the root is between 0.44 and 0.45.\r<BR>\n" );
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document.write( "Using a calculator, we find the root is approximately $x \approx 0.4468$.\r<BR>\n" );
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document.write( "Now, plug this into $f(x)$:\r<BR>\n" );
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document.write( "$$f(0.4468) \approx (0.4468)^3 + 2(0.4468)^2 + 3(0.4468) + 2 + \frac{1}{0.4468} \approx 0.089 + 0.399 + 1.340 + 2 + 2.238 \approx 6.066$$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{6}$<BR>\n" );
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