Inflection Point of a function
- Tolaso J Kos
- Administrator
- Posts: 867
- Joined: Sat Nov 07, 2015 6:12 pm
- Location: Larisa
- Contact:
Inflection Point of a function
Prove that the function \( f(x)=x^{x^{x}}-x^x \) has an inflection point at \(x=1 \).
I may be overlooking something here.. but I cannot prove that directly.
I may be overlooking something here.. but I cannot prove that directly.
Imagination is much more important than knowledge.
-
- Former Team Member
- Posts: 77
- Joined: Mon Nov 09, 2015 11:52 am
- Location: Limassol/Pyla Cyprus
- Contact:
Re: Inflection Point of a function
I presume you can show that \(f'(1) = 1\) showing that \(x=1\) is a stationary point. Now observe that for \(0 < x < 1\) we have \(x^x < x^1 = x\). So \(x^{x^x} < x^x\), i.e. \(f(x) < 0 = f(1).\) Also, for \(x > 1\) we have \(x^x > x\) giving \(f(x) > f(1)\). So \(x=1\) is a point of inflection.
- Tolaso J Kos
- Administrator
- Posts: 867
- Joined: Sat Nov 07, 2015 6:12 pm
- Location: Larisa
- Contact:
Re: Inflection Point of a function
Demetres I cannot follow your solution but I have another approach.
A bit tedious however.
Finding the second derivative (easy but a tedious task) we see that \( f''(1)=0 \).
In order to show that \( f \) has an inflection point at \( x=1 \) we're envoking a theorem that states:
Theorem: If \( g \) is differentiable , \(g(a)=0 \) and \(g'(a) \neq 0 \) then \(g \) changes sign at a neigbourhood of \(a \).
To complete the proof we have to prove that \( g'''(1) \neq 0 \) . Of course we are not computing the third derivative. Instead we are expanding the function at a Taylor series around \( x=1 \) and see that the coefficient of the third power is indeed non zero and we are done.
A bit tedious however.
Finding the second derivative (easy but a tedious task) we see that \( f''(1)=0 \).
In order to show that \( f \) has an inflection point at \( x=1 \) we're envoking a theorem that states:
Theorem: If \( g \) is differentiable , \(g(a)=0 \) and \(g'(a) \neq 0 \) then \(g \) changes sign at a neigbourhood of \(a \).
To complete the proof we have to prove that \( g'''(1) \neq 0 \) . Of course we are not computing the third derivative. Instead we are expanding the function at a Taylor series around \( x=1 \) and see that the coefficient of the third power is indeed non zero and we are done.
Imagination is much more important than knowledge.
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 15 guests