differentiation

[EspoirTron]

There is a point of inflexion at x if and only if f"(x)=0 at x. Since f"(x) is the derivative function of f'(x), then we can say that a point of inflexion in f(x) is a local min/max in the f'(x).

I would be very careful in making that assumption. Consider f(x)=x^4.

[lzxnl]

There is a point of inflexion at x if and only if f"(x)=0 at x. Since f"(x) is the derivative function of f'(x), then we can say that a point of inflexion in f(x) is a local min/max in the f'(x).

There is a point of inflection at x=a if and only if f''(x) changes sign at x=a (and, of course, f(x) is twice differentiable at x=a).
So, what you can say is that if f''(a)=0 and f'''(a) is not 0, x=a is a point of inflection. If f'''(a)=0, you'll need to examine higher order derivatives; if you keep differentiating and you keep getting zeros at x=a, if the first non-zero derivative after the second derivative is an even derivative, you have a stationary point (classified by the sign of this even derivative; positive means local minimum, negative means local maximum) and if the first non-zero derivative is an odd derivative, you have a stationary point of inflection. This assumes all higher order derivatives exist at x=a. Confused? Well you generally won't come across functions in VCE with first, second and third derivatives all zero.

[Bestie]

yes very confusing... but yup i got it...

Thank you

Mod Edit: Took out question that was answered elsewhere - Phy124