Question 1209599
.
The function D(x) = sin(|x|) is concave up on the interval

1. ]π, 2π[


2. ]-π, 0[


3. ]-4π, -3π[


4. ]-2π, -π[
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<pre>
D(x) = sin(|x|).


The plot of the function D(|x|) is the same as the plot of sin(x) for positive x >= 0.


The plot of the function D(|x|) at x <= 0 is the mirror reflection about y-axis 
of the plot of the function sin(x) at x >= 0.


In positive domain x >= 0,  function sin(x) has the plot concave up in intervals ({{{pi+2k*pi}}},{{{2pi + 2k*pi)}}}).


In particular, function sin(x) has the plot concave up in interval ({{{pi}}},{{{2pi)}}}), which is option (1).

Hence, function D(x) has the plot concave up in the interval ({{{pi}}},{{{2pi)}}}), which is option (1).




In the negative domain x <= 0, function D(x) = sin(|x|) has the plot concave up in the intervals 
({{{-2pi+2k*pi}}},{{{-pi + 2k*pi)}}}), that are mirror reflections of intervals ({{{pi+2k*pi}}},{{{2pi + 2k*pi)}}}).   (*)


In particular, function sin(|x|) has the plot concave up in interval ({{{-2pi}}},{{{-pi)}}}), which is option (4).

Hence, function D(x) has the plot concave up in the interval ({{{-2pi}}},{{{-pi)}}}), which is option (4).



But in interval ({{{-pi}}},{{{0}}}), function D(x) is NOT concave up (vice versa, it is concave down).

Hence, option (2) is not the solution.



In accordance with (*), function D(x) has the plot concave up in the interval ({{{-4pi}}},{{{-3pi)}}}), 
which is option (3).



So, the intervals where D(x) has the plot concave up are (1), (3), and (4).



<U>ANSWER</U>.  Choices (1), (3) and (4).
</pre>


See the plot of the function D(x) under this link


<A HREF=https://www.desmos.com/calculator/cmaszxpgs6>https://www.desmos.com/calculator/cmaszxpgs6</A>


https://www.desmos.com/calculator/cmaszxpgs6


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Solved.