document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194381>Question 1194381</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the following function in two variables x and y<BR>\n" );
document.write( "f(x, y) = x^3y+2x^4+y^5<BR>\n" );
document.write( "a. Find the Jacobean determinant at point P (3,2)<BR>\n" );
document.write( "b. Find the Hessian determinant at point P (1,3)<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 #848526 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=848526\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Find the Jacobian Determinant at point P (3, 2)**\r<BR>\n" );
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document.write( "1. **Calculate Partial Derivatives:**\r<BR>\n" );
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document.write( "   - ∂f/∂x = 3x²y + 8x³ <BR>\n" );
document.write( "   - ∂f/∂y = x³ + 5y⁴\r<BR>\n" );
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document.write( "2. **Evaluate Partial Derivatives at P (3, 2):**\r<BR>\n" );
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document.write( "   - ∂f/∂x(3, 2) = 3(3)²(2) + 8(3)³ = 54 + 216 = 270<BR>\n" );
document.write( "   - ∂f/∂y(3, 2) = (3)³ + 5(2)⁴ = 27 + 80 = 107\r<BR>\n" );
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document.write( "3. **Construct the Jacobian Matrix:**\r<BR>\n" );
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document.write( "   - The Jacobian matrix is a 1x2 matrix: <BR>\n" );
document.write( "      [ ∂f/∂x  ∂f/∂y ] <BR>\n" );
document.write( "      [ 270   107 ]\r<BR>\n" );
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document.write( "4. **Calculate the Jacobian Determinant:**\r<BR>\n" );
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document.write( "   - Since it's a 1x2 matrix, the determinant is not defined. The Jacobian determinant is only defined for square matrices (where the number of rows equals the number of columns).\r<BR>\n" );
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document.write( "**b. Find the Hessian Determinant at point P (1, 3)**\r<BR>\n" );
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document.write( "1. **Calculate Second-Order Partial Derivatives:**\r<BR>\n" );
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document.write( "   - ∂²f/∂x² = 6xy + 24x²<BR>\n" );
document.write( "   - ∂²f/∂y² = 20y³<BR>\n" );
document.write( "   - ∂²f/∂x∂y = 3x² <BR>\n" );
document.write( "   - ∂²f/∂y∂x = 3x² \r<BR>\n" );
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document.write( "2. **Evaluate Second-Order Partial Derivatives at P (1, 3):**\r<BR>\n" );
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document.write( "   - ∂²f/∂x²(1, 3) = 6(1)(3) + 24(1)² = 18 + 24 = 42<BR>\n" );
document.write( "   - ∂²f/∂y²(1, 3) = 20(3)³ = 540<BR>\n" );
document.write( "   - ∂²f/∂x∂y(1, 3) = 3(1)² = 3<BR>\n" );
document.write( "   - ∂²f/∂y∂x(1, 3) = 3(1)² = 3\r<BR>\n" );
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document.write( "3. **Construct the Hessian Matrix:**\r<BR>\n" );
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document.write( "   - The Hessian matrix is a 2x2 matrix:<BR>\n" );
document.write( "      [ ∂²f/∂x²   ∂²f/∂x∂y ]<BR>\n" );
document.write( "      [ ∂²f/∂y∂x   ∂²f/∂y² ]<BR>\n" );
document.write( "      [ 42        3      ]<BR>\n" );
document.write( "      [ 3        540     ]\r<BR>\n" );
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document.write( "4. **Calculate the Hessian Determinant:**\r<BR>\n" );
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document.write( "   - det(Hessian) = (42)(540) - (3)(3) = 22680 - 9 = 22671\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* The Jacobian determinant at point P (3, 2) is not defined.<BR>\n" );
document.write( "* The Hessian determinant at point P (1, 3) is 22671.<BR>\n" );
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