For each of the following equations, prove it can be solved for x as a $C^1$-function of y and z in a neighborhood of (0,0,0)

Question

For each of the following equations, prove it can be solved for x as a $C^1$-function of y and z in a neighborhood of (0,0,0). Have x,y,z in R.

(a) $cos(x)^2-e^{sin(xy)^3+x}=z^2$

(b) $(x^2+y^3+z^4)^2=sin(x-y+z)$

(c) $x^7+ye^zx^3-x^2+x=log(1+y^2+z^2)$

I'm sure if I knew how to do one of them, I would be able to figure the other two out. I believe it requires using the implicit function theorem, but I'm not sure sure how to apply it in the 3 variable case. Also notice(x,y,z)=(0,0,0) as a solution for all 3, which I believe is useful.

This is my first question here so hopefully I haven't messed anything up. Thanks everyone.

Answer 1

Welcome to Mathematics StackExchange!

You are correct in that you may use the implicit function theorem. In each case, bring everything to the left side of the equation and see that you are in a situation of the kind $$f(x,y,z) = 0$$ so these conditions identify a level surface of some function $f$. What regularity does $f$ have? You know that the origin is in this level surface for all three, so that can be the point mentioned in the IFT statement. Are all the hypotheses satisfied? (Hint: what about the regularity of the restriction of the jacobian of $f$ to the dependent variable, in this case $x$, evaluated at the origin?)

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Addendum. For completeness, here is the statement:

Theorem. Let $\Omega\subseteq \mathbb R^{n+m}$ be an open set and let $\mathbf f : \Omega \to \mathbb R^m$ be of class $C^k$. Suppose $\mathbf f$ vanishes in a point $(\mathbf x_0,\mathbf y_0)\in\Omega$, with $\mathbf x_0\in\mathbb R^n$ and $\mathbf y_0\in\mathbb R^m$, and suppose that the $m\times m$ restriction of the jacobian matrix to the $\mathbf y$ variables is non singular at that point. Then there exist $\Omega’\subseteq \mathbb R^n$ and $\Omega’’\in\mathbb R^m$, both open, such that

1. First of all, $(\mathbf x_0, \mathbf y_0)\in \Omega’\times\Omega’’\subseteq\Omega$;
2. For all $\mathbf x \in\Omega’$ there exists a unique $\mathbf y\in\Omega’’$ such that $\mathbf f(\mathbf x, \mathbf y) = \mathbf 0$;
3. The function $\varphi : \Omega’ \to \Omega’’$ that associates to each $\mathbf x$ such unique $\mathbf y$ is of class $C^k$.

In your case, $n=2$, $m=1$, and $k=1$, and the point in question is the origin.