Basic question: First derivative test to detect whether a function is decreasing

[seeker101]

If the first derivative of a function f from R to R is negative on [a,b], it IS right to say that the function is decreasing on [a,b] right?

Are there any other ways of showing that the function is decreasing on [a,b]?

 

[tiny-tim]

Hi seeker101!

If the first derivative of a function f from R to R is negative on [a,b], it IS right to say that the function is decreasing on [a,b] right?

Yes.

Are there any other ways of showing that the function is decreasing on [a,b]?

No … for a differentiable R->R function, negative derivative at a point is the same as decreasing.

 

[EnumaElish]

That's correct as a definition. But, what the OP is asking may be whether there are other ways of arguing that a function is decreasing over an interval.

 

[nicksauce]

Of course there are other ways to show a function is increasing (the same argument can be applied to decreasing) on an interval. For example, using the definition of increasing: if x,y are in [a,b] then show y > x implies f(y) >= f(x).

Example: Show that f(x) = x^2 is increasing on [0,2].
Let x and y belong to [0,2] with y > x. Then we can write y = x + e, for some e>0. Then f(y) = (x+e)^2 = x^2 + 2xe + e^2 > f(x) = x^2, since e^2 is >0 and 2xe is >0.

 

[qntty]

how about a function f is decreasing on [a,b] if (\forall x_1,x_2\in [a,b]) x_1<x_2 \implies f(x_1) \ge f(x_2)