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# Non-horizontal points of inflexion

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1. My textbook mentions non-horizontal points of inflexion but doesn't define them. Are they points
when f''(x)=0 but f'(x) doesn't =0? Whereas horizontal points of inflexion have f'(x)=0, i.e. are
stationary. Does anyone have an idea about what sort of question these would come up in during
A2-level equivalent exams?

Thanks, AR
2. In article <[email protected] om>, AR <[email protected]> wrote:
[q1]>My textbook mentions non-horizontal points of inflexion but doesn't define them. Are they points[/q1]
[q1]>when f''(x)=0 but f'(x) doesn't =0? Whereas horizontal[/q1]

Almost. They are points where f'' changes sign. So f'' must vanish at a point of inflexion, but not
all points where f'' vanishes have to be PI---for example, f(x)=x^4 has a local (and indeed global)
minimum at x=0, even though f''(0)=0.

[q1]>points of inflexion have f'(x)=0, i.e. are stationary. Does anyone have an idea about what sort of[/q1]
[q1]>question these would come up in during A2-level equivalent exams?[/q1]

Not sure: this sort of thing is required if you have to sketch graphs of functions. Whether that
comes up in those exams is outside my ken.

--
Rob. http://www.mis.coventry.ac.uk/~mtx014/
3. Originally posted by Ar
My textbook mentions non-horizontal points of inflexion but doesn't define them. Are they points
when f''(x)=0 but f'(x) doesn't =0? Whereas horizontal points of inflexion have f'(x)=0, i.e. are
stationary.

Thanks, AR
Using <> for 'doesn't equal',

at horizontal (stationary) points of inflection, f'(x)=0 and f''(x)=0,

at non-horizontal (non-stationary) points of inflection, f'(x)<>0 and f''(x)<>0,

and there is a further condition that for a point of inflection (regardless of whether it is stationary or non-stationary), either f'''(x) or some even higher derivative of x must be nonzero.

Otherwise for example one would deduce a point of inflection at x=0 for y=x^6. This would be incorrect because in fact there is a minimum there. With this particular function the first derivative to be non-zero at x=0 is the sixth ).
4. Originally posted by James P

Using <> for 'doesn't equal',

at horizontal (stationary) points of inflection, f'(x)=0 and f''(x)=0,

at non-horizontal (non-stationary) points of inflection, f'(x)<>0 and f''(x)<>0,

and there is a further condition that for a point of inflection (regardless of whether it is stationary or non-stationary), either f'''(x) or some even higher derivative of x must be nonzero.

Otherwise for example one would deduce a point of inflection at x=0 for y=x^6. This would be incorrect because in fact there is a minimum there. With this particular function the first derivative to be non-zero at x=0 is the sixth ).
Whoops! For a point of inflection the 'further condition' is simply that f'''(x) must be nonzero.

James P

Updated: June 1, 2002
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