Question

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

Answer 1

`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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