We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$ a) Let $m \in \mathbb{R}$. Show that $T_{m}$ is of class $C^{1}$ on $\mathbb{R}_{+}^{*}$, and calculate $T_{m}^{\prime}$ in terms of $T_{m-1}$. b) Let $m \in \mathbb{R}$. Is the function $T_{m}$ of class $C^{\infty}$ on $\mathbb{R}_{+}^{*}$? What is the monotonicity of $T_{m}$ on $\mathbb{R}_{+}^{*}$? Is the function $T_{m}$ convex on $\mathbb{R}_{+}^{*}$? c) Discuss as a function of $m \in \mathbb{R}$ the right-differentiability of $T_{m}$ at 0.
We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$
$$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $m \in \mathbb{R}$. Show that $T_{m}$ is of class $C^{1}$ on $\mathbb{R}_{+}^{*}$, and calculate $T_{m}^{\prime}$ in terms of $T_{m-1}$.
b) Let $m \in \mathbb{R}$. Is the function $T_{m}$ of class $C^{\infty}$ on $\mathbb{R}_{+}^{*}$? What is the monotonicity of $T_{m}$ on $\mathbb{R}_{+}^{*}$? Is the function $T_{m}$ convex on $\mathbb{R}_{+}^{*}$?
c) Discuss as a function of $m \in \mathbb{R}$ the right-differentiability of $T_{m}$ at 0.