MHB Derivative of Constant: Proving f(x) is Constant

[Suvadip]

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 ?

 

[MarkFL]

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?

 

[Ackbach]

Re: derivative of constant

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:
$$\Delta y=f'(x)\, \Delta x = 0 \cdot \Delta x \, ...$$

 

[HallsofIvy]

Re: derivative of constant

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).