Let $g : \mathbb{R}_+ \rightarrow \mathbb{R}$ be the function defined by $g(x) = \ln\left(1 - p + pe^x\right)$ for all $x \geq 0$. a. Show that $g$ is well defined and of class $C^2$ on $\mathbb{R}_+$. For $x \geq 0$, express $g''(x)$ in the form $\frac{\alpha\beta}{(\alpha+\beta)^2}$ where $\alpha$ and $\beta$ are positive reals that may depend on $x$. b. Show that $g''(x) \leq \frac{1}{4}$ for all $x \geq 0$. c. Show that $$\ln\left(1 - p + pe^x\right) \leq px + \frac{x^2}{8} \text{ for all } x \geq 0$$
Let $g : \mathbb{R}_+ \rightarrow \mathbb{R}$ be the function defined by $g(x) = \ln\left(1 - p + pe^x\right)$ for all $x \geq 0$.
a. Show that $g$ is well defined and of class $C^2$ on $\mathbb{R}_+$. For $x \geq 0$, express $g''(x)$ in the form $\frac{\alpha\beta}{(\alpha+\beta)^2}$ where $\alpha$ and $\beta$ are positive reals that may depend on $x$.
b. Show that $g''(x) \leq \frac{1}{4}$ for all $x \geq 0$.
c. Show that
$$\ln\left(1 - p + pe^x\right) \leq px + \frac{x^2}{8} \text{ for all } x \geq 0$$