Chapter 8.2, Problem 36E

Let $f\left(x\right)$ and $g\left(x\right)$ be analytic at $x_0$. Determine whether the following statements are always true or sometimes false:

a. $3f\left(x\right)+g\left(x\right)$ is analytic at $x_0$

b. $f\left(x\right)/g\left(x\right)$ is analytic at $x_0$

c. $f^{\prime }\left(x\right)$ is analytic at $x_0$

d. ${\left[f\left(x\right)\right]}^3-{\displaystyle {\int }_{x_0}^{x}g\left(t\right)dt}$ is analytic at $x_0$