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\n" ); document.write( "Question:
\n" ); document.write( "Let d,e and g be real numbers and consider the function F(x) = (x-d)(x-e)(x-g). Using Calculus, find the x-coordinate of the point of the inflection for the given function.
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\n" ); document.write( "Solution:
\n" ); document.write( "Given a function F(x) differentiable around x=x0 where x0 locates a point of inflection.
\n" ); document.write( "A point of inflection located at x=x0 has the property that F\"(x0)=0, which means that this is only a necessary condition.
\n" ); document.write( "A sufficient condition is that F\"(x) must have different signs on either side of x0.
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\n" ); document.write( "With the above information, we will search for the point of inflection of the function F(x)=(x-d)(x-e)(x-g).
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\n" ); document.write( "- Differentiate F(x) twice to get
\n" ); document.write( "F'(x)=(x-e)(x-g)+(x-d)(x-g)+(x-d)(x-e), and
\n" ); document.write( "F\"(x)=6x-2g-2e-2d
\n" ); document.write( "- Since F\"(x)=0 is a necessary condition for an inflection point, we solve for x in F\"(x)=0 to get
\n" ); document.write( "x0=(d+e+g)/3.
\n" ); document.write( "- We need to check if the point is in fact a point of inflection using the sufficient condition:
\n" ); document.write( "- Let ε= a very small number, then x0+ε and x0-ε are on each side of x0.
\n" ); document.write( "- We calculate
\n" ); document.write( "Q=F\"(x0+ε)*F\"(x0-ε)
\n" ); document.write( "=[+6ε][-6ε]
\n" ); document.write( "=-36ε²
\n" ); document.write( "<0 (less than zero, therefore satisfies the sufficient condition)
\n" ); document.write( "- Hence x0=(d+e+g)/3 is an inflection point.
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