(1) We know cosx £ 1 for all x. By considering the derivative of the function
f(x) = x - sinx
prove that sinx £ x for x ³ 0.

(2) By considering the derivative of the function
f(x) = cosx - (1 - x2
2
 )
prove that
cosx ³ 1 - x2
2

for x ³ 0.

(3) By considering the derivative of the function
f(x) = (x - x3
3!
 ) - sinx
prove that
sinx ³ (x - x3
3!
 )

for x ³ 0.

(4) By considering the derivative of the function
f(x) = cosx - (1 - x2
2!
 + x4
4!
 )
prove that
cosx £ (1 - x2
2!
 + x4
4!
 )

for x ³ 0.

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