document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209869>Question 1209869</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the value of x that maximizes<BR>\n" );
document.write( "f(x) = \log (-20x + 16 \sqrt{x} - x). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850924 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=850924\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Understand the Domain**\r<BR>\n" );
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document.write( "First, we need to find the domain of the function inside the logarithm. We need:\r<BR>\n" );
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document.write( "* `-20x + 16√x - x > 0`<BR>\n" );
document.write( "* `-21x + 16√x > 0`<BR>\n" );
document.write( "* `16√x > 21x`\r<BR>\n" );
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document.write( "Since `√x` and `x` are involved, we know that `x ≥ 0`.\r<BR>\n" );
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document.write( "Let `y = √x`, so `x = y²`. Then the inequality becomes:\r<BR>\n" );
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document.write( "* `16y > 21y²`<BR>\n" );
document.write( "* `16y - 21y² > 0`<BR>\n" );
document.write( "* `y(16 - 21y) > 0`\r<BR>\n" );
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document.write( "This inequality holds when `0 < y < 16/21`. Since `y = √x`, we have:\r<BR>\n" );
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document.write( "* `0 < √x < 16/21`<BR>\n" );
document.write( "* `0 < x < (16/21)²`<BR>\n" );
document.write( "* `0 < x < 256/441`\r<BR>\n" );
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document.write( "So the domain is `0 < x < 256/441`.\r<BR>\n" );
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document.write( "**2. Maximize the Inside of the Logarithm**\r<BR>\n" );
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document.write( "Since the logarithm is an increasing function, maximizing `f(x)` is equivalent to maximizing the expression inside the logarithm:\r<BR>\n" );
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document.write( "* `g(x) = -21x + 16√x`\r<BR>\n" );
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document.write( "Let `y = √x` again. Then `g(x) = -21y² + 16y`.\r<BR>\n" );
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document.write( "This is a quadratic function in `y`. To find its maximum, we can complete the square or find the vertex.\r<BR>\n" );
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document.write( "The vertex of a quadratic `ay² + by + c` is at `y = -b / (2a)`. In our case:\r<BR>\n" );
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document.write( "* `y = -16 / (2 * -21) = 16 / 42 = 8 / 21`\r<BR>\n" );
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document.write( "Now, substitute back `y = √x`:\r<BR>\n" );
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document.write( "* `√x = 8 / 21`<BR>\n" );
document.write( "* `x = (8 / 21)² = 64 / 441`\r<BR>\n" );
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document.write( "Since `64/441` is within the domain `(0, 256/441)`, this is a valid maximum.\r<BR>\n" );
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document.write( "**3. Verify the Maximum**\r<BR>\n" );
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document.write( "To ensure this is a maximum, we can take the second derivative of `g(x)` with respect to `x`:\r<BR>\n" );
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document.write( "* `g(x) = -21x + 16x^(1/2)`<BR>\n" );
document.write( "* `g'(x) = -21 + 8x^(-1/2)`<BR>\n" );
document.write( "* `g''(x) = -4x^(-3/2)`\r<BR>\n" );
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document.write( "Since `g''(x)` is negative for all `x > 0`, the function `g(x)` is concave down, and the value we found is indeed a maximum.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "The value of `x` that maximizes `f(x) = log(-20x + 16√x - x)` is:\r<BR>\n" );
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document.write( "* `x = 64 / 441`<BR>\n" );
document.write( " </SPAN>\n" );
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