How Do You Prove the Range of a Square Root Polynomial Function?

[mtayab1994]

Homework Statement

f:[1,+∞[→ℝ
x→sqrt(x+2)-sqrt(x-1)

Homework Equations

show that f([1,+∞[)=]0,sqrt(3)]

The Attempt at a Solution

Any tips on how to start it.

 

[Ray Vickson]

Homework Statement

f:[1,+∞[→ℝ
x→sqrt(x+2)-sqrt(x-1)

Homework Equations

show that f([1,+∞[)=]0,sqrt(3))

The Attempt at a Solution

Any tips on how to start it.

What have you done so far?

RGV

 

[mtayab1994]

What have you done so far?

RGV

well nothing really, i want something that'll help me get going.

 

[Deveno]

things to try:

1. calculate f(x) explicitly for a few values of x. i always like 0,1,-1 and 42 (ok, 0,1 and-1 won't work. pick something else. maybe 2,4 and 6)

2. see if lim x→∞ and lim x→1+ exist.

3. see if f(x) has a global maximum or minimum (yeah, derivatives, we can use them, right?)

 

[mtayab1994]

things to try:

1. calculate f(x) explicitly for a few values of x. i always like 0,1,-1 and 42 (ok, 0,1 and-1 won't work. pick something else. maybe 2,4 and 6)

2. see if lim x→∞ and lim x→1+ exist.

3. see if f(x) has a global maximum or minimum (yeah, derivatives, we can use them, right?)

alright so these 3 steps should help me?

 

[Deveno]

you want to figure out how f(x) behaves. you need to get some information.

 

[mtayab1994]

yea but in class when we want to solve something like this he have to show that f(x)=y or f(x)≥0.

 

[Deveno]

what is f(1)?

we need to establish "some" things about f. it's going to be impossible to do, if all you think is "well, f is a function".

take it's derivative. it is always positive, does it ever equal 0, is it some places positive, and some places negative? what does that tell you about f?

are there any places where f(x) = 0 (does this have anything to do with whether or not f crosses the x-axis)?

what happens when x gets "really really big"? does it have a limit as x→∞? if so, what is this limit?