Derivative of Constant: Proving f(x) is Constant

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Discussion Overview

The discussion revolves around the relationship between the derivative of a function and the constancy of that function. Participants explore how to prove that if the derivative of a function \( f(x) \) is zero, then the function itself must be constant. The scope includes theoretical reasoning and mathematical proofs related to derivatives and constants.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant states that if \( f(x) \) is constant, then \( f'(x) = 0 \), but questions how to prove the reverse implication that \( f'(x) = 0 \) implies \( f(x) \) is constant.
  • Another participant suggests considering the definition of the derivative and proposes integrating with respect to the independent variable as a potential method.
  • A third participant reiterates the initial claim and introduces the concept of differentials, stating that if \( f'(x) = 0 \), then \( \Delta y = f'(x) \Delta x = 0 \cdot \Delta x \).
  • A fourth participant references the mean value theorem, indicating that if \( f'(x) = 0 \) for all \( x \), then for any two points \( a \) and \( b \), it follows that \( f(a) = f(b) \), suggesting this as a way to show that \( f(x) \) is constant.

Areas of Agreement / Disagreement

Participants express similar initial claims regarding the relationship between the derivative and constancy, but there is no consensus on a definitive proof for the reverse implication. Multiple approaches are suggested, indicating ongoing exploration and debate.

Contextual Notes

Some participants' arguments depend on the assumptions of continuity and differentiability of the function \( f(x) \), which are not explicitly stated or agreed upon in the discussion.

Suvadip
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If $$f(x)$$ is constant then I can show that $$f'(x)=0$$.
But if $$f'(x)=0$$, then how to show that $$f(x)$$ is constant ?
 
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Re: derivative of constant

What does the definition of the derivative tell you?

edit: Sorry, I misread...have you tried integrating with respect to the independent variable?
 
Last edited:
Re: derivative of constant

suvadip said:
If $$f(x)$$ is constant then I can show that $$f'(x)=0$$.
But if $$f'(x)=0$$, then how to show that $$f(x)$$ is constant ?

Another approach would be to use differentials:
$$\Delta y=f'(x)\, \Delta x = 0 \cdot \Delta x \, ...$$
 
Re: derivative of constant

suvadip said:
If $$f(x)$$ is constant then I can show that $$f'(x)=0$$.
But if $$f'(x)=0$$, then how to show that $$f(x)$$ is constant ?
The mean value theorem: if f is continuous and differentiable on [a, b] then there exist c in [a, b] such that f'(c)= \frac{f(b)- f(a)}{b- a}. If f'(x)= 0 for all x, then for any a and b, \frac{f(b)- f(a)}{b- a}= 0[/tex] from which f(a)= f(b).
 

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