# Real analysis

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29.12 (a) Show that x < tan x for all x in (0, π/2).

(b) Show that x/ sin x is strictly increasing function on (0, π/2).

(c) Show that x ≤ (π/2)ּsin x for all x in [0, π/2].

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#### Solution Preview

Please see the attachment.

We need to following theorems for the questions.

Theorem 1: Suppose the function is differentiable on the interval . If for all , then is strictly increasing on . If for all , then is strictly decreasing on . If for all , then is increasing on . If for all , then is decreasing on .

Theorem 2 (Quotient Rule): Suppose the functions ...

#### Solution Summary

There are three proofs here involving trigonometric functions.

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