document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195392>Question 1195392</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sketch the graph of a single function that has all of the properties listed.<BR>\n" );
document.write( "a. Continuous and differentiable for all real numbers<BR>\n" );
document.write( "b. f'(x)<0 on (-∞,-4) and(0,5)<BR>\n" );
document.write( "c. f'(x)>0 on (-4,0) and (5,∞)<BR>\n" );
document.write( "d. f''(x)>0 on (-∞,-1) and (2,∞)<BR>\n" );
document.write( "e. f''(x)<0 on (-1,2)<BR>\n" );
document.write( "f. f'(-4)=f'(5)=0<BR>\n" );
document.write( "g.f''(x)=0 at (-1,11) and (2,10)<BR>\n" );
document.write( "any help will be so appreciative this is the only question left on my homework that is stumping me<BR>\n" );
document.write( "thank you so much in advance! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848431 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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=848431\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's sketch a function that satisfies all the given properties.\r<BR>\n" );
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document.write( "**1. Understand the Properties**\r<BR>\n" );
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document.write( "* **a. Continuous and differentiable for all real numbers:** This means the graph has no breaks or sharp corners.<BR>\n" );
document.write( "* **b. f'(x) < 0 on (-∞,-4) and (0,5):** The function is decreasing on these intervals (slope is negative).<BR>\n" );
document.write( "* **c. f'(x) > 0 on (-4,0) and (5,∞):** The function is increasing on these intervals (slope is positive).<BR>\n" );
document.write( "* **d. f''(x) > 0 on (-∞,-1) and (2,∞):** The function is concave up (opens upwards) on these intervals.<BR>\n" );
document.write( "* **e. f''(x) < 0 on (-1,2):** The function is concave down (opens downwards) on this interval.<BR>\n" );
document.write( "* **f. f'(-4) = f'(5) = 0:** There are horizontal tangents (critical points) at x = -4 and x = 5.<BR>\n" );
document.write( "* **g. f''(x) = 0 at (-1,11) and (2,10):** There are inflection points at (-1,11) and (2,10) where concavity changes.\r<BR>\n" );
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document.write( "**2. Sketch the Graph**\r<BR>\n" );
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document.write( "* **Start with the concavity:**<BR>\n" );
document.write( "    * Concave up on (-∞, -1) and (2, ∞)<BR>\n" );
document.write( "    * Concave down on (-1, 2)\r<BR>\n" );
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document.write( "* **Add the critical points:**<BR>\n" );
document.write( "    * Horizontal tangents at x = -4 and x = 5\r<BR>\n" );
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document.write( "* **Determine increasing/decreasing intervals:**<BR>\n" );
document.write( "    * Decreasing on (-∞, -4) and (0, 5)<BR>\n" );
document.write( "    * Increasing on (-4, 0) and (5, ∞)\r<BR>\n" );
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document.write( "* **Connect the points smoothly:**<BR>\n" );
document.write( "    * Ensure the graph is continuous and differentiable everywhere.<BR>\n" );
document.write( "    * Make sure the graph reflects the concavity and increasing/decreasing behavior.\r<BR>\n" );
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document.write( "**Here's a rough sketch of a possible function:**\r<BR>\n" );
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document.write( "* **(Note: This is just one possible representation. There could be variations that still satisfy all the given properties.)**\r<BR>\n" );
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document.write( "   * The graph would start by increasing and concave up from (-∞, -4).<BR>\n" );
document.write( "   * At x = -4, it would have a horizontal tangent and continue increasing, but now concave down.<BR>\n" );
document.write( "   * At x = -1, there's an inflection point (concavity changes).<BR>\n" );
document.write( "   * The graph continues increasing and concave down until x = 0.<BR>\n" );
document.write( "   * At x = 0, it has another horizontal tangent and starts decreasing.<BR>\n" );
document.write( "   * At x = 2, there's another inflection point (concavity changes).<BR>\n" );
document.write( "   * The graph continues decreasing and concave up until x = 5.<BR>\n" );
document.write( "   * At x = 5, it has a horizontal tangent and starts increasing and remaining concave up towards positive infinity.\r<BR>\n" );
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document.write( "**Key Points:**\r<BR>\n" );
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document.write( "* The graph should have the general shape described above, reflecting the given properties of the function's derivatives.<BR>\n" );
document.write( "* The exact positions and heights of the points are not fixed, as long as they satisfy the given conditions.\r<BR>\n" );
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document.write( "I hope this helps! Let me know if you'd like to explore specific aspects or have further questions.<BR>\n" );
document.write( " </SPAN>\n" );
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