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I need your help to determine all $a,b,c \in \mathbb{R}$ for which mapping $f:\mathbb{R} \to \mathbb{R}$, $$f(x)=ax^3+bx^2+cx$$ is a Diffeomorphism.

Majid
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Melina
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  • Well, let's recall the definition of a diffeomorphism:

    Def: Let A,B be differentiable manifolds (i.e. second countable topological spaces that are locally homeomorphic to $\mathbb R^n$). Then a diffeomorphism is a differentiable bijection f:$A\rightarrow B$ whose inverse is also differentiable.

    This is pretty easy in this case since A = B = $\mathbb R$ and polynomials are clearly differentiable on subsets of $\mathbb R$. What makes it tricky is whether or not f is a bijection is determined by a,b,c.

     – Mathemagician1234 Jul 30 '16 at 01:02
  • @Mathemagician1234 $f'(x)>0$ or $<0$ seems to be a sufficient and necessary condition. – Vim Jul 30 '16 at 01:04

2 Answers2

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$f'(x)=3ax^2+2bx+c$, $\Delta=4b^2-12ac$. $\Delta<0$ is i.e $b^2<3ac$, $f$ is strictly monotone, since $lim_{x\rightarrow -\infty}f(x)=-\infty$ if $a>0, +\infty$ if $a<0$ and $lim_{x\rightarrow +\infty}f(x)=+\infty$ if $a>0, -\infty$ if $a>0$, $f$ is bijective and locally invertible thus is a diffeomorphim.

$\Delta\geq 0$ implies there exists $x$ such that $f'(x)=0$, so $f$ is not a diffeomorphism since the differential of a diffeomorphism must be invertible.

Tsemo Aristide
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A differentiable function $f: \Bbb R \to \Bbb R$ is a diffeomorphism if and only if:

$(1)$ $f'(x) > 0$ for all $x$, or $f'(x) < 0$ for all $x$.

$(2)$ $\lim_{x \to - \infty} f(x) = -\infty$ and $\lim_{x \to + \infty }f(x) = + \infty$, or $\lim_{x \to - \infty} f(x) = +\infty$ and $\lim_{x \to + \infty }f(x) = - \infty$

I leave the proof of this as an exercise for you.

Suppose that $a = 0$. If $b \neq 0$, then clearly $f$ is not a diffeomorphism. If $b =0$, then $f$ is a diffeomorphism iff $c \neq 0$.

Suppose that $a \neq 0$. We have $f'(x) = 3ax^2 + 2bx + c$. So condition $(1)$ is satisfied iff $b^2 - 3ac < 0$. Now, supposing that, we have that $f$ is a cubic polynomial, so it clearly satisfies $(2)$.

Therefore, $f$ is a diffeomorphism iff:

  • $a = 0$, $b = 0$ and $c \neq 0$

or

  • $a \neq 0$ and $b^2 < 3ac$.
  • Your first sentence is false. Your two conditions say nothing about differentiability. – user281392 Jul 30 '16 at 01:19
  • It is still false. Consider $f(x)=x^3$. – user281392 Jul 30 '16 at 01:20
  • I think "strictly monotone" is not what you mean in the last paragraph. The condition you write is equivalent to having a strictly positive or strictly negative derivative. – user281392 Jul 30 '16 at 01:25
  • @user281392 thank you for pointing out the trivial and non-trivial errors. Tip: when an answer contains many errors, even though useful, the best thing to do is to downvote it. This saves everybody's time and forces the answerer to completely revise their answer. –  Jul 30 '16 at 01:38