Let I be an open interval contained in the domain of a real valued function f. Then f is said to be increasing on I if a<b Rightarrow f(a) leq f(b) for all a, b in I. And a theorem given after it is f is increasing in I if f′(x)>0 forall x in I. Shouldn't it be f′(x) geq 0.

crazygbyo 2022-08-12 Answered
Confusion between the definition of increasing function and a theorem regarding it.
The definition of increasing function given in my school maths text book is
Let I be an open interval contained in the domain of a real valued function f. Then f is said to be increasing on I if a < b f ( a ) f ( b ) for all a , b I.
And a theorem given after it is
f is increasing in I if f ( x ) > 0 x I.
Shouldn't it be f ( x ) 0.
Is constant function an increasing function? Or a function like f ( x ) = x 3 where f ( x ) = 0 at some or all points which are increasing according to definition.
Edit: The definition of decreasing function given is
f is decreasing on I if a < b f ( a ) f ( b ).
A constant function follow this definition too.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Gaige Burton
Answered 2022-08-13 Author has 16 answers
Step 1
Some would call x I , f ( x ) 0 a non decreasing function.
Also some would call x I , f ( x ) > 0 a strictly increasing function.
But, yes, x I , f ( x ) 0 is often defined as an increasing function. In that case, the function f ( x ) = 1 is an increasing function.
Sometime the word monotone gets thrown in there too.
You may as well get used to the fact that not all mathematicians use the same definition for things. As long as the book is consistent, that is acceptable.
Step 2
INCREASING:
- x , y I , x < y f ( x ) f ( y )
- x I , f ( x ) 0
STRICTLY INCREASING:
- x , y I , x < y f ( x ) < f ( y )
- x I , f ( x ) > 0
MOTONIC:
- Strictly increasing or strictly decreasing
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-19
Stability, critical points and similar properties of solutions of nonlinear Volterra integral equations
I have a system of nonlinear Volterra integral equations of form x ( t ) = x 0 + 0 t K ( t , s ) F ( x ( s ) ) d s and I am interested on the critical points of x(t), I mean maximum, minimum, increasing and decreasing intervals, nonnegativity etc.
I imagine it's impossible to get complete informations about that, but here I am asking for theorems and general results to help me to study these aspects, once is impossible know the true solution.
asked 2022-08-10
Find the rate of change of main dependent variable
We have f : R R , f ( x ) = x 2 + x sin ( x ), and we need to find intervals of monotonicity. Here is all my steps:
f ( x ) = 2 x + x cos ( x ) + sin ( x )
f ( x ) = 0 x = 0 the only solution.
Now I need to find where f′ is positive and negative. I don't want to put value for f′ to find the sign, I want another method. So I tried to differentiate the function again to see if f′ is increasing or decreasing:
f ( x ) = 2 x sin ( x ) + cos ( x ), but I don't know if f ( x ) 0 or f ( x ) 0.
How can I find the sign for f′ to determine monotony of f ?
asked 2022-08-20
How to find where the function is decreasing/increasing/concave/convex f ( x ) = 2 1 + x 2 ?
I need to find where this function is increasing, decreasing, concave and convex. I've found it's derivative:
f ( x ) = 4 x ( 1 + x 2 ) 2
Now you're supposed to make either f ( x ) > 0 when it's increasing and f ( x ) < 0 when it's decreasing, but that gives:
Increasing: x < 0. Decreasing: x > 0.
But what does that actually mean? It's just confusing, usually when I solve these you get 2 solutions, so it's for example increasing on the interval of (-2,2). What does this one tell me? What's the easiest way to find where this function is increasing and decreasing?
Then I also did the second derivative, which is:
f ( x ) = 4 ( 3 x 2 1 ) ( 1 + x 2 ) 3
How does this all help me find my solution?
asked 2022-07-18
Question on Local Maxima and Local Minima
Find the set of all the possible values of a for which the function f ( x ) = 5 + ( a 2 ) x + ( a 1 ) x 2 x 3 has a local minimum value at some x < 1 and local maximum value at some x > 1.
The first derivative of f(x) is:
f ( x ) = ( a 2 ) + 2 x ( a 1 ) 3 x 2
I do know the first derivative test for local maxima and local minima, but I can't figure out how I could use monotonicity to find intervals of increase and decrease of f′(x)
The expression for f′(x) might suggest the double derivative test is the key, considering f ( x ) = 2 ( a 1 ) 6 x for which the intervals where it is greater than zero and less than zero can be easily found, but then again I can't think of a way how I could find a c such that f ( c ) = 0.
asked 2022-08-05
Determine convergence / divergence of sin π n 2 .
Let a n = sin π n 2 .
I attempted the integral test but on the interval [ 1 , 2 ) it is increasing and decreasing on ( 2 , ). So the integral test in only applicable for the decreasing part. + the integral computation seems to lead to 3 pages of steps...
I believe the comparison test would be the most reasonable test, graphically I observed that a n behaves like b n = 1 / n when n is large.
b n = ; but, since b n > a n inconclusive for divergence/convergence.
I attempted the limit comparison, and ratio test, but inconclusive. I am uncertain if I am doing them properly.
How could I bound below a n to proceed with the comparison test? Is there a more appropriate method? How would you proceed?
asked 2022-08-17
Maximmum and minimum values of function in interval
How to find the intervals ( Increasing and decreasing) of the function and its Maximum and minimum value, where the function: f ( x ) = a x 2 + b x + c.
asked 2022-08-09
How can one determinate the variationt of f(g(x))
Given the two functions:
f ( x ) = x 2 2 x
g ( x ) = x + 1
The question is determinate the variation of f(g(x))
We have D f ( g ( x ) ) = ( 1 , + )
For f it's decreasing for x < 1.
Increasing for x > 1
For g its increasing for x > 1
To determinate the variationf of f(g(x))
In interval ( 1 ; + )
We have g is increasing
X > 1 means that g ( x ) > g ( 1 ) so g ( x ) > 0.
So g ( [ 1 ; + ) ) = [ 0 , + ).
But the problem is that f in that interval is increasing and decreasing Im stuck here.
Can one write a methode to answer any question like this theoricaly.