Prove that $c:\mathbb{R}\rightarrow\mathbb{R}^2$ with $t\rightarrow (t^2,t^3)$ is not regular.

Question

Prove that $c:\mathbb{R}\rightarrow\mathbb{R}^2$ with $t\rightarrow (t^2,t^3)$ is not regular.

Ok so this is probably a well-known problem of Differential Geometry but I have problem understanding it.Any online source explaining it in details will be helpful.

Otherwise I am trying to show the proof of the class notes. I am giving the images below as I think that if I write it here people might think that I have made some typing mistakes or stuff.

Answer 1

The range of $c$ is $\{(x,x^{3/2}): x\ge 0\}.$ Let's call the range $c^*.$ Suppose $\gamma (t) =(f(t),g(t))$ is a $C^1$ map of $\mathbb R$ onto $c^*.$ Then $f$ maps $\mathbb R$ surjectively onto $[0,\infty).$ Hence $f(t_0) = 0$ for some $t_0.$ This implies $f$ has a minimum at $t_0,$ which tells us $f'(t_0) = 0.$ Now $g(t)=f(t)^{3/2}$ for all $t.$ Thus $g'(t_0) = (3/2)f(t_0)^{1/2}f'(t_0) = 0.$ Therefore $\gamma'(t_0) = (0,0),$ proving $\gamma $ is not regular.