Maximum value of y

[Vemuri]

If k is a positive constant and y = |x - k| - |x + k|, what is the maximum value of y?

(1) x < 0
(2) k = 3

[mike22629]

Im not sure I am right.

IMO B

If that is right I will explain my answer.

[karmayogi]

IMO B.

There are two ways to solve the problem.
1. By observation
y = |x - k| - |x + k|
For k = +ve

y< 0 for x > k
y>0 for x< k
The value will be maximum when |x+k| = 0. Therefore for x = -k, y = 2k, which is the maximum value.

2. Above is not a fool proof method, but a quick one. The best solution is drawing graph.

First draw the graph of y = x - k and y = x + k
Second draw the graph of y = |x-k| and y = |x+k|
Second graph will give you some idea. You can solve the problem from here itself. However, we can go step ahead and draw the actual graph of

y = |x - k| - |x + k|

The graph is attached with the post. Given value of K we can find the maximum value of y.

PS: you can also draw the final graph directly by plotting the values of x and assuming some constant value for k.

[mike22629]

Well I guess he explained it for you. By the way thanks for showing the graphing way to do it. Much easier.

Thanks Karmayogi!