document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164224>Question 1164224</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the values of a and b such that the following function is differentiable at 0\r<BR>\n" );
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document.write( "f(x) = ax3cos(1/x)+bx +b , if x < 0<BR>\n" );
document.write( "       sqrt(a+bx), if x>= 0\r<BR>\n" );
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document.write( "From what I know, I need to prove that the function is continuous at x=0 thus limf(x) as x approaches 0 must exist and must be equals to f(0) which led me to the equation b = sqrt(a) limf(x) as x approaches 0 from left and from right. But from there I do not know how to proceed. Please advise. Thank you very much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #788654 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=788654\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "In order to be differentiable at a point, the function must be both continuous at that point and the derivative must exist at that point.\r<BR>\n" );
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document.write( "In order to be continuous at a point, the limit from the left must equal the limit from the right.\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right{0^-}}\,\[ax^3\cos\(\frac{1}{x}\)\,+\,bx\,+\,b\]\ =\ b\">\r<BR>\n" );
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document.write( "must be equal to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right{0^+}}\,\[\sqrt{a\,+\,bx}]\ =\ \sqrt{a}\">\r<BR>\n" );
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document.write( "So the first condition is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ \sqrt{a}\">\r<BR>\n" );
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document.write( "But the derivative must exist at zero as well, so using the definition of the derivative and the function value at zero from either side as determined by evaluation of the above limits, we have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right{0^-}}\,\[\frac{\(ax^3\cos\frac{1}{x}\,+\,bx\,+\,b\)\ -\ b}{x}\]\ =\ b\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right{0^+}}\,\[\frac{\sqrt{a\,+\,bx}\,-\,\sqrt{a}}{x}\]\ =\ \frac{b}{2\sqrt{a}}\">\r<BR>\n" );
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document.write( "And that gives us the second condition: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ \frac{b}{2\sqrt{a}}\">\r<BR>\n" );
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document.write( "Substituting:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ b\ =\ \frac{b}{2b}\ \Right\ b\ =\ \frac{1}{2}\">\r<BR>\n" );
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document.write( "Then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ b^2\ \Right\ a\ =\ \frac{1}{4}\">\r<BR>\n" );
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document.write( "																<BR>\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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