# 1617-257/TUT-R-7

Suppose $F : \mathbb{R}^2 \to \mathbb{R}^2$ is $C^1$ with $F(t^4, e^{t^2}) = (4,5)$ for all $t \in \mathbb{R}$. Prove that $DF(0,1)$ is not invertible.
Let $\gamma(t) := (t^4, e^{t^2})$. At the end of the tutorial, a couple of students pointed out that it's also true that $DF(\gamma(t))$ is not invertible for any $t \in \mathbb{R}$. It's easy to check that $\gamma'(t) \neq 0$ when $t \neq 0$ and that $\gamma'(t)$ is in the kernel of $DF(\gamma(t))$.
It takes some more care to prove what the problem is asking because one can't immediately deduce that $DF(\gamma(t))$ has non-trivial kernel with $\gamma'(0) = 0$. A key observation here is that since $F$ is $C^1$, invertibility of $DF$ at $\gamma(0)$ implies invertibility of $DF$ at a neighborhood of $\gamma(0)$.