Sum of monotonic increasing and monotonic decreasing functionsConsider an interval x ∈ [ x 0...

Awainaideannagi

Awainaideannagi

Answered

2022-07-16

Sum of monotonic increasing and monotonic decreasing functions
Consider 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.

Answer & Explanation

eyiliweyouc

eyiliweyouc

Expert

2022-07-17Added 15 answers

Step 1
Alas, 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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?