First slide

Continuity

Question

The set of points of continuity of the function $y = \sqrt{\frac{1}{2} - \cos^{2} x}$ contains in the interval

Difficult

A

$[\frac{π}{4}, \frac{3 π}{4}]$
B

$[\frac{5 π}{4}, \frac{7 π}{4}]$
C

$[\frac{21 π}{4}, \frac{23 π}{4}]$
D

All the above

Solution

$y = \sqrt{\frac{1}{2} - \cos^{2} x}$
If is defined when $\frac{1}{2} - \cos^{2} x \geq 0$
$\Rightarrow \frac{1}{2} \geq \cos^{2} x$
i.e., $0 \leq \cos^{2} x \leq \frac{1}{2}$
$\Rightarrow - \frac{1}{\sqrt{2}} \leq \cos x \leq 0, 0 \leq \cos x \leq \frac{1}{\sqrt{2}}$
$\Rightarrow - \frac{1}{\sqrt{2}} \leq \cos x \leq \frac{1}{\sqrt{2}}$
$\Rightarrow x \in [\frac{π}{4}, \frac{3 π}{4}], [\frac{5 π}{4}, \frac{7 π}{4}], [\frac{21 π}{4}, \frac{23 π}{4}]$
$\Rightarrow x \in [2 n π - \frac{3 π}{4}, 2 n π - \frac{π}{4}] \cup [2 n π + \frac{π}{4}, 2 n π + \frac{3 π}{4}], n \in z$
From all above

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