document.write( "Question 284312: the electronic potential at a point (x;y) on the same line segment extending from (0;3) to (2;0) is given by $\"P=3x%5E2%2B2y%5E2\"$ at what point on this line segment is the potentail a minimum? \n" ); document.write( "

Algebra.Com's Answer #206350 by Fombitz(32388) $\"\"$ $\"About$

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Find the line connecting (0,3) and (2,0).
\n" ); document.write( " $\"m=%280-3%29%2F%282-0%29=-3%2F2\"$
\n" ); document.write( " $\"y=mx%2Bb=-%283%2F2%29x%2Bb\"$
\n" ); document.write( "When x=0, y=3,
\n" ); document.write( " $\"y=-%283%2F2%29%280%29%2Bb=3\"$
\n" ); document.write( " $\"b=3\"$
\n" ); document.write( "So then
\n" ); document.write( " $\"y=-%283%2F2%29x%2B3\"$
\n" ); document.write( "Find $\"y%5E2\"$
\n" ); document.write( " $\"y%5E2=%28-%283%2F2%29x%2B3%29%5E2=%289%2F4%29x%5E2-9x%2B9\"$
\n" ); document.write( " $\"2y%5E2=%289%2F2%29x%5E2-18x%2B18\"$
\n" ); document.write( "Substitute into P(x,y),
\n" ); document.write( " $\"P%28x%2Cy%29=3x%5E2%2B2y%5E2=3x%5E2%2B%289%2F2%29x%5E2-18x%2B18\"$
\n" ); document.write( " $\"P%28x%29=%2815%2F2%29x%5E2-18x%2B18\"$
\n" ); document.write( "Now P is only a function of x.
\n" ); document.write( "Differentiate wrt x and set the derivative equal to zero to find the minimum.
\n" ); document.write( " $\"dP%2Fdx=%2815%2F2%29%282x%29-18=15x-18=0\"$
\n" ); document.write( " $\"15x=18\"$
\n" ); document.write( " $\"x=18%2F15=6%2F5\"$
\n" ); document.write( "Now going back to the line equation,
\n" ); document.write( " $\"y=-%283%2F2%29x%2B3\"$
\n" ); document.write( " $\"y=-%283%2F2%29%286%2F5%29%2B3=-9%2F5%2B15%2F5=6%2F5\"$
\n" ); document.write( "The minimum occurs at (6/5,6/5). \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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