# Questions tagged [implicit-function-theorem]

The implicit function theorem gives sufficient conditions to solve a given equation for one or more of the variables as functions of the remaining variables. The basic form of the theorem is that of an existence theorem. However, the contraction mapping proof of the theorem provides an error estimate for a sequence of approximating maps. Sometimes it is also termed the implicit mapping theorem. See http://en.wikipedia.org/wiki/Implicit_function_theorem

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### Implicit Function Theorem with $x^2+y^2+z^2=\psi(ax+by+cz)$

Given the equation $x^2+y^2+z^2=\psi(ax+by+cz)$, with $a,b,c\in\mathbb{R},\ c\neq 0$, and $\psi:\mathbb{R}\to\mathbb{R}$ that satisfies $\psi\in C^2,\ \psi(0)=0,\ \psi'(0)\neq0$, prove that in a ...
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### There exists a one-to-one function $g$ such that $f\circ g$ is constant [duplicate]

Let $f:\mathbb{R}^2\to \mathbb{R}$ be a $C^1$ function. Then there exists a continuous one-to-one function $g$ on $[0,1]$ such that $f\circ g=$constsnt. Attempt: If $f\equiv 0$ then nothing to prove. ...
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### $f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^1$, $f'(x) \neq 0$, show that $f$ is a $C^1$-isomorphism between $\mathbb{R}$ and $f(\mathbb{R})$

My question is just as above, and it was given as an exercise at the end of a section about the implicit function theorem & some similar theorems, but after inverse mapping theorem. By $C^1$-...
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### Prove that $xy^2+y^3z^4+z^5x^6=1$ has a solution in an open neighborhood about the point $x_0=(0,1,-1)$

Prove with implicit function theorem that $xy^2+y^3z^4+z^5x^6=1$ has a $C^1$ solution with a form of $(x,g(x,z),z)$ in an open neighborhood about the point $x_0=(0,1,-1)$. What I have gotten so far: ...
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### How to sketch an equation with variables x and y

I was doing a sample exam for my final tommorow and I did not understand how to solve the following question. Maybe it's relevant to note that during this math course we have been talking about the ...
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### Application of the Implicit function theorem on the function $f(x,y,z)=(x^2+y^2)e^z+\sin(\pi x)yz+2z-1$

I wanted to know if my procedure about this Implicit function theorem question is correct and if my results are correct. Note : I am an engineering student, so I'm mostly concerned about the "...