Let $U$ be the open set of $\mathbb{R}^{2}$ defined by $$U := \left\{(t, x) \in \mathbb{R}^{2} \mid t > 0 \text{ and } x > -t\right\}$$ and let $f$ be the function defined on $U$ by $$f(t, x) = t^{2} \ln\left(1 + \frac{x}{t}\right) - tx.$$ (a) Show that for $(t, x) \in U$, we have $$x \leqslant 0 \Rightarrow f(t, x) \leqslant -\frac{x^{2}}{2}.$$ (b) For $x > 0$, show that we have $$\forall t \geqslant 1, \quad f(t, x) \leqslant f(1, x).$$ For this, one may begin by writing $\frac{\partial f}{\partial t}(t, x)$ in the form $t F(x/t)$ for a certain function $F$ that one will study.
Let $U$ be the open set of $\mathbb{R}^{2}$ defined by
$$U := \left\{(t, x) \in \mathbb{R}^{2} \mid t > 0 \text{ and } x > -t\right\}$$
and let $f$ be the function defined on $U$ by
$$f(t, x) = t^{2} \ln\left(1 + \frac{x}{t}\right) - tx.$$
(a) Show that for $(t, x) \in U$, we have
$$x \leqslant 0 \Rightarrow f(t, x) \leqslant -\frac{x^{2}}{2}.$$
(b) For $x > 0$, show that we have
$$\forall t \geqslant 1, \quad f(t, x) \leqslant f(1, x).$$
For this, one may begin by writing $\frac{\partial f}{\partial t}(t, x)$ in the form $t F(x/t)$ for a certain function $F$ that one will study.