f(x)=√(x+1)-√(x) : show that f is decreasing

[akramzine]

Hello.I came across this question in my homework that I couldn't solve:
f(x)=sqrt(x+1)-sqrt(x)
show that f is decreasing. I tried to suppose that x>y
and show that f(x)-f(y)<0 but it didn't lead to anything.I would be grateful for any given help

 

[SAMUELK]

Hello.I came across this question in my homework that I couldn't solve:
f(x)=sqrt(x+1)-sqrt(x)
show that f is decreasing. I tried to suppose that x>y
and show that f(x)-f(y)<0 but it didn't lead to anything.I would be grateful for any given help

Any time you have sqrt()-sqrt(), there is a nice word that should pop into your head.

RATIONALIZE!

f(x)=sqrt(x+1)-sqrt(x)

1
= -------------------
sqrt(x+1) + sqrt(x)


f(x+h) = sqrt(x+h+1)-sqrt(x+h)
1
= ----------------------------
sqrt(x+h+1) + sqrt(x+h)

Now, which is bigger? Take it from there.

 

[akramzine]

thank you for the hint samuel I solved it