First slide

Functions (XII)

Question

The range of the following function is $f (x) = \sqrt{(1 - \cos x) \sqrt{(1 - \cos x) \sqrt{(1 - \cos x) \sqrt{. . . . \infty}}}}$

Moderate

A

$[0, 1]$
B

$[0, \frac{1}{2}]$
C

$[0, 2]$
D

None of these

Solution

Given $f (x) = \sqrt{(1 - \cos x) \sqrt{(1 - \cos x) \sqrt{(1 - \cos x) \sqrt{. . . . \infty}}}}$
$\begin{array}{l} = {(1 - \cos x)}^{\frac{1}{2}} {(1 - \cos x)}^{\frac{1}{4}} {(1 - - \cos x)}^{\frac{1}{8}} . . . . . \infty \\ = {(1 - \cos x)}^{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + . . . . . . \infty} \\ = {(1 - \cos x)}^{\frac{1 / 2}{1 - (1 / 2)}} \\ = 1 - \cos x \end{array}$
Thus, the range of f(x) is [0, 2].

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