In the "User's guide to viscosity solutions" by Crandall, Ishii and Lions, they proposed the following notion:
Let $\varphi:\mathbb{R}^n\rightarrow\mathbb{R}$ and $\bar{x}\in\mathbb{R}^n$. The superjet of $\varphi$ at $\bar{x}$ is defined by $$ \small J^{2,+}\varphi(\bar{x}):=\left\{(p,A)\in\mathbb{R}^n\times \mathbb{S}^n:\limsup_{x\rightarrow\bar{x}}\frac{\varphi(x)-\varphi(\bar{x})-\langle x^*,x-\bar{x}\rangle-\frac{1}{2}\langle A(x-\bar{x}),x-\bar{x}\rangle}{\|x-\bar{x}\|^2}\leq 0\right\}, $$ where $\mathbb{S}^n$ is the set of symmetric $n\times n$ matrices.
They made the following claim as an interesting exercise: $$ J^{2,+}\varphi(\bar{x})=\left\{(\nabla s(\bar{x}),\nabla^2s(\bar{x})):s \;\text{is}\;C^2\;\text{and}\;\varphi-s\;\text{has a local maximum at}\;\bar{x}\right\}. $$ I have tried without success to prove this claim and would appreciate some help. Moreover I would like to ask whether or not $$ J^{2,+}\varphi(\bar{x})=\left\{(\nabla s(\bar{x}),\nabla^2s(\bar{x})):s \;\text{is convex,}\;C^2\;\text{and}\;\varphi-s\;\text{has a local maximum at}\;\bar{x}\right\}. $$