SOLUTION: If f(x) is a linear function such that f(2) ≤ f(3), f(4) ≥ f(5), and f(6)=10. Prove that line "f" is a flat line.

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Question 1036056: If f(x) is a linear function such that f(2) ≤ f(3), f(4) ≥ f(5), and f(6)=10.
Prove that line "f" is a flat line.
Found 2 solutions by robertb, Edwin McCravy:
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
f(x) a linear function implies that f(x) = ax + b for some constants a and b.
Note that a is the slope of the line and b is the y-intercept of the line.
Now f%282%29+=+2a+%2B+b+%3C=+3a+%2B+b+=+f%283%29 ==> 0+%3C=+a.
Also,
f%284%29+=+4a+%2B+b+%3E=+5a+%2B+b+=+f%285%29 ==> 0+%3E=+a.
Hence 0+%3C=+a and 0+%3E=+a ==> a = 0 ==> line has slope 0, and thus it is a horizontal line, or a "flat" line.
==> f(x) = b, and since f(6) = 10 = b, the linear function is highlight%28f%28x%29+=+10%29.

Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
f(2) ≤ f(3), f(4) ≥ f(5), and f(6)=10

Since f(x) is a linear function, there exist numbers m
and b such that f(x) = mx+b

   2m+b ≤ 3m+b,      and 4m+b ≥ 5m+b,  and 6m+b = 10
                                              b = 10-6m

Substituting 10-6m for b

2m+10-6m ≤ 3m+10-6m, and 4m+10-6m ≥ 5m+10-6m

  -4m+10 ≤ -3m+10,   and   -2m+10 ≥ -m+b

      -m ≤ 0,        and       -m ≥ 0

       m ≥ 0         and        m ≤ 0

The only way both can be true is for slope m = 0 

Flat lines have slope 0, and ONLY flat lines
have slope 0.  Thus "f" is a flat line.

Edwin