(a) Let $g:[0,1] \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ be a continuous function such that for each $t \in[0,1]$, the partial derivatives $D_{i} g(t, x)(i=1, \ldots, n)$ of $x \mapsto g(t, x)$ exist and are continuous on $[0,1] \times \mathbb{R}^{n}$. Define $G: \mathbb{R}^{n} \rightarrow \mathbb{R}$ by

$$G(x)=\int_{0}^{1} g(t, x) d t$$

Show that $G$ has continuous partial derivatives $D_{i} G$ given by

$$D_{i} G(x)=\int_{0}^{1} D_{i} g(t, x) d t$$

for $i=1, \ldots, n$.

(b) Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be an infinitely differentiable function, that is, partial derivatives $D_{i_{1}} D_{i_{2}} \cdots D_{i_{k}} f$ exist and are continuous for all $k \in \mathbb{N}$ and $i_{1}, \ldots, i_{k} \in\{1,2\}$. Show that for any $\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}$,

$$f\left(x_{1}, x_{2}\right)=f\left(x_{1}, 0\right)+x_{2} D_{2} f\left(x_{1}, 0\right)+x_{2}^{2} h\left(x_{1}, x_{2}\right)$$

where $h: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is an infinitely differentiable function.

[Hint: You may use the fact that if $u: \mathbb{R} \rightarrow \mathbb{R}$ is infinitely differentiable, then

$$\left.u(1)=u(0)+u^{\prime}(0)+\int_{0}^{1}(1-t) u^{\prime \prime}(t) d t .\right]$$