What are the points of inflection of #f(x)=x^7/(4x-2) #?

1 Answer
Jul 16, 2016

#x=(35+-sqrt 217)/48#.
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Explanation:

#(4x-2)f=x^7#. Differentiate twice..

#f'(4x-2)+4f=7x^6#

#f''(4x-2)+8f'=42x^5#

Eliminating f' and f,
#f''=(42x^5-8(7x^6-(4x^7)/(4x-2)))/(4x-2)#

f''=0, for x satisfying #42x^5-8(7x^6-(4x^7)/(4x-2))=0#. This leads to

#x=0 and (42-56x)(4x-2)+32x^2=0#-

The second reduces to #48x^2-70x+21=9# giving #x=(35+-sqrt 217)/48#.

Now, use the first three equations..

At x=0, all derivatives vanish.

At x=(35+-sqrt 217)/48#, f''' is not 0

So, the points of inflexion are .#x=(35+-sqrt 217)/48#.

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