Hit me. Twice. Doing these at work is ALWAYS a bad idea.
However, the lines along which I was thinking:
If x, y and z all > 1, then obviously, 1 and 2 are sufficient.
Also, making x or y or z < -1 would not help much since powers of 2 and 4 would be positive and keep increasing anyway.
So, 0 to 1 interval sounds like the one we need to check.
Now, if we have 0<x<1, y = let's say 2 and z just slightly larger than 2 so as to satisfy statement 1, can we have 4th power increase enough to compensate.
x=1/2, y=2, z=2.01 (or in other words 2+1/100)
x+y = 2 + 1/2 = 2.5 > z
Statement 2 is satisfied.
x^2 + y^2 = 4 + 1/16
z^2 = (2+1/100)^2 = 4 + 2*2*1/100 + 0.01*0.01 = 4 + 0.04 + 0.0001
Do the comparison
4 + 1/16 ? 4 + 0.0401
1/16 ? 0.0401
0.0625 > 0.401
So, statement 1 is satisfied.
Now, x^4 + y^4 = 1/16 + 16
z^4 = (z^2)^2 = (4 + 0.0401)^2 = 16 + 8*0.0401 + 0.0016...
Do the comparison
1/16 + 16 ? 16 + 8*0.0401 + 0.0016
0.0625 ? 0.32 + 0.0016
0.0625 < 0.32 + 0.0016
So, x^4 + y^4 < z^4
Now, these calculations seem lengthy and tedious but write it out on a piece of paper and they seem to fly. Still, it took me little over 2 min. to do it on paper. (The earlier post was done all on notepad
) but I figured this one deserved a notebook and pencil which I try to avoid at work for obvious reasons!)
The gist is, to get y (or x) and z real close to each other and buffer out x (has to < 1) such that x^2 compensates the difference between z^2 - y^2 but x^4 does not compensate the diff between z^4 - y^4.
If I came across this during real test, I would've guessed D or E and moved on depending on how much time I had left compared to how many questions left and how much progress I was able to make under real time test conditions.
And if there is some other mistake here, I will refrain from doing these at work in future.
