How do i show that y decreasing function for x>2?

Question

it is given..
when y=x/x-2 , x cannot equals 2

i found dy/dx to be -2/(x-2)square
to ans the above question, what working should i show?
is saying..
"when x>2, dy/dx will always be negative so this proves that y is a decreasing function"
enough?

Answer 1

Yes. Showing that the 1'st derivative is negative over the domain of interest is enough to prove that the function is everywhere decreasing.

You do *not* need any 2'nd derivative tests and, in particular, you can't have a 1'st or 2'nd derivative at a point where the function fails to exist (in this case at x=2)

Firat_c knows what he's talking about. Marios_a doesn't have a clue. Even if you get a zero for the 2'nd derivative it doesn't mean you've screwed up. It means that the point in question is what's called an 'inflection point'. Don't worry, you'll get to those later.


Doug

Answer 2

-2/(x-2)^2 is a negative number except for x=2 when it is not defined. And as you said having the derivative negative is enough to show that the function you have is decreasing.

By the way Marios's answer below is wrong, don't get confused. I think he/she is probably thinking about the second derivative test which is used for determining wether a critical point is a local max or min. It does not help with the inc/dec property of a function.

Answer 3

You don't need calculus for this! Suppose that 2-2x', and therefore x(x'-2)>x'(x-2). Divide both sides by x-2 and x'-2, which are positive, and you get

x'/x'-2 < x/x-2. So as x grows, y gets smaller, which is to say that this is a decreasing function. You don't need derivatives or limits at all.

Answer 4

find the second derivative(i.e. d(sqr)y/dx(sqr) ) and substitute where x=2. then if the result is negative then the fuction is increasing. if it's positive then it's decreasing. if equal to zero then the method fails i.e. u've done some mistake in ur calculus.

Answer 5

it might help to justify "will always be negative" when x>2
although it's clearly true that when a negative numerator is divided by a positive denominator, the result is negative ...

but your reasoning seems sound, just a bit disorganized

Answer 6

thats enough it think