document.write( "Question 1145198: Find k and the points T of contact if the parabola y=x^2 - k touches the cubic y=x^3 \n" ); document.write( "

Algebra.Com's Answer #766415 by Alan3354(69443) $\"\"$ $\"About$

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Find k and the points T of contact if the parabola y=x^2 - k touches the cubic y=x^3
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\n" ); document.write( "If by \"touch\" you mean tangent:
\n" ); document.write( "If k = 0, they are tangent at the Origin, and there is an intersection.
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\n" ); document.write( "y = x^3
\n" ); document.write( "y' = 3x^2 (1st derivative = slope at x)
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\n" ); document.write( "y = x^2 - k
\n" ); document.write( "y' = 2x
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\n" ); document.write( "Find where the 2 slopes are equal.
\n" ); document.write( "3x^2 = 2x
\n" ); document.write( "3x^2 - 2x = 0
\n" ); document.write( "x = 0
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\n" ); document.write( "3x = 2
\n" ); document.write( "x = 2/3
\n" ); document.write( "---
\n" ); document.write( "(2/3)^3 - (2/3)^2 + k = 0
\n" ); document.write( "8/27 - 12/27 = -k
\n" ); document.write( "k = 4/27
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\n" ); document.write( "And there's an intersection in Q3.
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