Can tip this problem. I did a part , but could not complete. They gave me a tip to complete , but could not. I thank the help .
$\mathbf{Problem}$ Let $\, \, \, f: \mathbb{S}^{n+1} \rightarrow \mathbb{R}$ analytic, $ n \geq 1 $. Consider the non-singular connected manifold $V_t = f^{-1}(t)$, $t \in \mathbb{R}$ and $t_0$ denote a critical value of $f$. Let $\mathrm{inj}(V_t)$ denote the injectivity radius of $V_t$. Then $$ \mathrm{inj}(V_t) \geq C \, \vert t-t_0 \vert^{\gamma}, \quad \gamma > 0.$$
$\mathbf{Proof:}$ We first note that on $V_t$ we have in suitable local coordinates $f(x_1,...,x_{n+1}) = t$. Assume that $x_{n+1}$ is the normal to the surface $V_t$ at $z_0$. Thefore, follows by Lojasicwiz inequality
\begin{align} \left\vert \frac{\partial f}{\partial x_{n+1}} (z_0) \right\vert = \left\vert \nabla f(z_0) \right\vert \geq C \, \left\vert t - t_0 \right\vert^{\alpha}. \end{align}
Observe that there is a neighbourhood of $z_0$ of radius $C \vert t-t_0 \vert^\alpha$ such that for all $z$ in this neighborhood we alsi have
\begin{align} \left\vert \frac{\partial f}{\partial x_{n+1}} (z) \right\vert \geq C \, \left\vert t-t_0 \right\vert^{\alpha} \end{align}
Indeed, note that
\begin{align} C \, \left\vert t-t_0 \right\vert^{\alpha} \leq \left\vert \frac{\partial f}{\partial x_{n+1}} (z_0) \right\vert &\leq \left\vert \frac{\partial f}{\partial x_{n+1}} (z_0) - \frac{\partial f}{\partial x_{n+1}} (z) \right\vert + \left\vert \frac{\partial f}{\partial x_{n+1}} (z) \right\vert \\ &\leq C \, \left\vert z-z_0 \right\vert + \left\vert \frac{\partial f}{\partial x_{n+1}} (z) \right\vert \end{align} once $f \in \mathrm{C}^2.$ Thus for $\left\vert z-z_0 \right\vert \leq \frac{C}{2} \, \left\vert t-t_0 \right\vert^\alpha$, have $$\left\vert \frac{\partial f}{\partial x_{n+1}} (z) \right\vert \geq C \, \left\vert t-t_0 \right\vert^\alpha $$ for all z for which $\left\vert z-z_0 \right\vert \leq \left\vert t-t_0 \right\vert^{\alpha} $. Thus the implicit function theorem will tell us that $V_t$ is given by a graph in a neighborhood of $z_0$ of radius $C \, \left\vert t-t_0 \right\vert.$
HINT FOR CONCLUSION:
The fact that $\left\vert \frac{\partial f}{\partial x_{n+1}} (z) \right\vert \geq C \, \left\vert t-t_0 \right\vert^{\alpha}$ for all a for wich $\left\vert z-z_0 \right\vert \geq \left\vert t-t_0 \right\vert^{\alpha}$ implies that the absolute value of the dot product of the graph at $z$ and the unit vector along the positive $x_{n+1}$ axis is bounded below by $C \, \left\vert t-t_0 \right\vert^{\alpha}$ for all $z$ for which $\left\vert z-z_0 \right\vert \leq C \, \left\vert t-t_0 \right\vert^{\alpha}$. Thus by a projection onto the tangent plane of $V_t$ at $z_0$, the problem follows in fact with the choice $\gamma = 2\alpha .$