Suppose f:R-IR is a diffeetiable function which satısfies flo) = 0 We wart to show f': IR → IR is a coustant fanction if and cnly if fl*+yl =fx9 + fiy) for all ×, y € IR
Suppose f:R-IR is a diffeetiable function which satısfies flo) = 0 We wart to show f': IR → IR is a coustant fanction if and cnly if fl*+yl =fx9 + fiy) for all ×, y € IR
Intermediate Algebra
10th Edition
ISBN:9781285195728
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Functions
Section9.1: Relations And Functions
Problem 76PS
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![Suppose f.R→IR is a diffetinble funtion which satısfies flo) = 0
We want to show f': IR → IR is a coustant function if and anly if flx+yl =frx + fiy) for all ×, y & IR](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d01de64-689e-49bb-980b-df7bc0b532f7%2F427dfa25-798c-4ad3-a54d-8db09125aede%2Fn5hjo3n_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose f.R→IR is a diffetinble funtion which satısfies flo) = 0
We want to show f': IR → IR is a coustant function if and anly if flx+yl =frx + fiy) for all ×, y & IR
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
Step 1
Given that is a differentiable function which satisfies .
It is required to show that is constant function if and only if for all .
Suppose if is constant function.
Then it is required to prove that for all .
Since is constant function, for all .
Integrating on both sides with respect to x will give
...... (1)
where is the integration constant.
Given that the function satisfies .
Substituting x=0 in (1) will give
Thus the integration constant .
Then (1) becomes
Now for all ,
Hence proved that if is constant function, then for all . .......... (A)
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