(1) We know

\cos x \leq 1

for all

x

. By considering the derivative of the function

f (x) = x - \sin x

prove that

\sin x \leq x

for

x \geq 0

(2) By considering the derivative of the function

$f (x) = \cos x - (1 - \frac{x^{2}}{2})$

prove that $\cos x \geq 1 - \frac{x^{2}}{2}$ for $x \geq 0$ .

(3) By considering the derivative of the function

$f (x) = (x - \frac{x^{3}}{3!}) - \sin x$

prove that $\sin x \geq (x - \frac{x^{3}}{3!})$ for $x \geq 0$ .

(4) By considering the derivative of the function

$f (x) = \cos x - (1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!})$

prove that $\cos x \leq (1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!})$ for $x \geq 0$ .

(5) What can you say about continuing this process?