D(x) = sin(|x|)

Concave upward means like a U. 

Absolute values are always positive or zero.  So in D(x), we are always taking
sines of positive or zero numbers, so D(x) will only be concave upward when the
sines of positive numbers or zero are negative or zero.  This only occurs in
quadrants 3 and 4. 

All choices except 2.

Answer: choices 1, 3, and 4.



Edwin

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 (,).


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

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




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


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

Hence, function D(x) has the plot concave up in the interval (,), which is option (4).



But in interval (,), 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 (,), 
which is option (3).



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



ANSWER.  Choices (1), (3) and (4).