document.write( "Question 1192427: Given that y = √4x-7 - 2/3(x-4), find dy/dx and show that d^y/dx^2 = -4/(4x-7)√4x-7 . Hence find the maximum point of the curve. \n" ); document.write( "

Algebra.Com's Answer #824481 by greenestamps(13198) $\"\"$ $\"About$

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\n" ); document.write( "Note it is not good form to use the \"√\" symbol when writing your function -- we have to guess how much of the following expression is under the radical. Use \"sqrt\" with parentheses around the radicand.

\n" ); document.write( " $\"y=sqrt%284x-7%29-%282%2F3%29%28x-4%29=%284x-7%29%5E%281%2F2%29-%282%2F3%29%28x-4%29\"$

\n" ); document.write( " $\"dy%2Fdx=%281%2F2%29%284%29%284x-7%29%5E%28-1%2F2%29-2%2F3=2%2Fsqrt%284x-7%29-2%2F3\"$

\n" ); document.write( "The maximum or minimum point(s) are where the derivative is zero.

\n" ); document.write( " $\"2%2Fsqrt%284x-7%29-2%2F3=0\"$
\n" ); document.write( " $\"2%2Fsqrt%284x-7%29=2%2F3\"$
\n" ); document.write( " $\"sqrt%284x-7%29=3\"$
\n" ); document.write( " $\"4x-7=9\"$
\n" ); document.write( " $\"4x=16\"$
\n" ); document.write( " $\"x=4\"$

\n" ); document.write( "There is a single maximum or minimum, at x=4.

\n" ); document.write( " $\"dy%2Fdx=2%284x-7%29%5E%28-1%2F2%29-2%2F3\"$
\n" ); document.write( "

\n" ); document.write( "The second derivative is always negative, so the graph has a local maximum at x=4.

\n" ); document.write( " $\"f%284%29=sqrt%284x-7%29-%282%2F3%29%28x-4%29=sqrt%2816-7%29-%282%2F3%29%280%29=sqrt%289%29=3\"$

\n" ); document.write( "ANSWER: The maximum point on the curve is (4,3)

\n" ); document.write( "A graph, showing (not very well) the maximum point at (4,3)

\n" ); document.write( " $\"graph%28400%2C400%2C-1%2C7%2C-1%2C5%2Csqrt%284x-7%29-%282%2F3%29%28x-4%29%29\"$

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