Properties of functions

Exercise

1.

Show that $f(x) = x - \tan{x}$ is strictly monotonically decreasing function on $${(-\frac{\pi}{2}, \frac{\pi}{2})}$$．

2.

Prove the following inequality．

(a)

For $x > 0$， $${\frac{x}{1+x} < \log{(1+x)}}$$

(b)

For $x > 0$， $${\frac{x}{1+x^{2}} < \tan^{-1}{x} < x}$$

(c)

$${e^{\pi} > \pi^{e}}$$

3.

Find the local extremum and concavity of the following function.

(a)

$${f(x) = x^{3} - 6x^2 + 9x + 3}$$

(b)

$${f(x) = x^{2}e^{-x}}$$