Sum of monotonic increasing and monotonic decreasing functionsConsider an interval x ∈ [ x 0 , x 1 ]. Assume there are two functions f(x) and g(x) with f ′ ( x ) ≥ 0 and g ′ ( x ) ≤ 0. We know that f ( x 0 ) ≤ 0, f ( x 1 ) ≥ 0, but g ( x ) ≥ 0 for all x ∈ [ x 0 , x 1 ]. I want to show that q ( x ) ≡ f ( x ) + g ( x ) will cross zero only once. We know that q ( x 0 ) ≤ 0 and q ( x 1 ) ≥ 0.

Question

Sum of monotonic increasing and monotonic decreasing functionsConsider an interval   x  ∈  [      x    0    ,      x    1    ]. Assume there are two functions f(x) and g(x) with       f    ′    (  x  )  ≥  0 and       g    ′    (  x  )  ≤  0. We know that   f  (      x    0    )  ≤  0,   f  (      x    1    )  ≥  0, but   g  (  x  )  ≥  0 for all   x  ∈  [      x    0    ,      x    1    ]. I want to show that   q  (  x  )  ≡  f  (  x  )  +  g  (  x  ) will cross zero only once. We know that   q  (      x    0    )  ≤  0 and   q  (      x    1    )  ≥  0.

eyiliweyouc · Accepted Answer

Step 1Alas, the answer is no.  f  (  x  )  =      {                            −          4                          x          ∈          [          0          ,          2          ]                                      −          2                          x          ∈          [          2          ,          4          ]                                      0                          x          ∈          [          4          ,          6          ]                          g  (  x  )  =      {                            5                          x          ∈          [          0          ,          1          ]                                      3                          x          ∈          [          1          ,          3          ]                                      1                          x          ∈          [          3          ,          5          ]                                      0                          x          ∈          [          5          ,          6          ]                        Step 2  q  (  x  )  =      {                            1                          x          ∈          [          0          ,          1          ]                                      −          1                          x          ∈          [          1          ,          2          ]                                      1                          x          ∈          [          2          ,          3          ]                                      −          1                          x          ∈          [          3          ,          4          ]                                      1                          x          ∈          [          4          ,          5          ]                                      0                          x          ∈          [          5          ,          6          ]                        This example could be made continuous and strictly monotone with some tweaking.