Given y=v (x/4-x) , complete the following.?

Question

I usually know how to work these out if they have a square root on top or just on the bottom, but this one happens to be over the entire function and it confuses me to no end.

So the following function

b. Determine y-intercepts (0)
----This one we just set x= 0 right?

c. determine x-intercepts (0)
----This one you just set y=0 right?

d. State domain of Function (-∞, 4) ; {x|x<4} ??
---- Work this one out please ? I'm kinda understanding this one. Kinda.

e. Write equations for all vertical asymptotes (Undefined x=4??)
---- For this one we have to do -+∞ of the lim of the original right? then it's -+4? which is undefined?

f. Determine the end behavior of the function (Not a polynomial so.. is that what you put or how do you work this out?)
----Explain? g. Use y'= 2 / x^1/2 (4-x) ^3/2 to determine the intervals where y is increasing
----Work this out please?

h. Use y'= 2 / x^1/2 (4-x) ^3/2 to determine the intervals where y is decreasing
----Work this one out please?

i. Give the coordinates, both x and y, of the relative maximum, if any. (apparently there is no max..
----Work this one out please?

j. Give the coordinates, both x and y, of the relative minimum, if any. (it's a min)
----Work this one out please?

k. Use y''= 4(x-1)/x^3/2(4-x)^5/2 to determine the open intervals where y is concave up (no concave up since it's all min)
----Work this one out please?

i. Use y''= 4(x-1)/x^3/2(4-x)^5/2 to determine the open intervals where y is concave down. (it's concave down)
----Work this one out please?

m. Give the coordinates of inflections points, if any. (No inflection points, how do you work this out? )
----Work this one out please?

I have the answers, but I want to make sure if I'm doing it right (as in working it out correctly-- that square root makes me feel like I'm messing it up. )

G-I is mainly the ones I want to see worked out.

Answer 1

y = √(x / (4-x))

 

a. Don't know, you didn't list a part a.

 

b. Determine y-intercepts (0)
---This one we just set x= 0 right?

Yes.
(0,0)

c. determine x-intercepts (0)
----This one you just set y=0 right?

Yes, though solving for x can sometimes be somewhat difficult.

(0,0)

d. State domain of Function (-∞, 4) ; {x|x<4} ??
---- Work this one out please ? I'm kinda understanding this one. Kinda.

The domain of a square root function is restricted by the fact that the expression under the radical cannot be negative.

In this case,

x/(4-x) ≥ 0

For this to be true, the numerator and denominator are either both negative or both positive.

x ≥ 0 AND 4-x ≥ 0

x ≥ 0 AND 4 ≥ x

0 ≤ x ≤ 4

-OR-

x ≤ 0 AND 4-x ≤ 0

x ≤ 0 AND 4 ≤ x

These cannot both happen at the same time.

 

Also, we cannot divide by zero, so

4-x ≠ 0

x ≠ 4

 

Combine these, and the domain is {x | 0 ≤ x < 4}, [0,4)

 

I don't see how the domain you quoted is correct. If you plug in -5, for example, y = √( -5 / (4 - -5) ) = √( -5/9 ) which is imaginary, and not in the domain. If I have written the function wrong, then the method still applies, you just need to solve different inequalities.

e. Write equations for all vertical asymptotes (Undefined x=4??)
---- For this one we have to do -+∞ of the lim of the original right? then it's -+4? which is undefined?

The vertical asymptote is x=4 here, because the denominator = 0 there. This is not the same as the infinite limits of the function. lim_x→±∞ would give you the horizontal asymptotes of the function, if any.

f. Determine the end behavior of the function (Not a polynomial so.. is that what you put or how do you work this out?)
----Explain?

To determine end behavior, take the limit of the function approaching the limits of your domain. In this case, that will be as x→0⁺ and x→4^-

lim_x→0+ √(x / (4-x)) = 0

lim_x→4- √(x / (4-x)) = +∞

 

g. Use y'= 2 / x^1/2 (4-x) ^3/2 to determine the intervals where y is increasing
----Work this out please?

To find critical points, you find where your derivative = 0. In this case, the derivative cannot be zero, so there are no critical points. Plug in any number in your domain to determine if it's increasing or decreasing.

y'(1) = 2 / [ 1^1/2 (4-1)^3/2 ]

I don't know (or care, really) what this number is, but I know it's positive, so the function is increasing everywhere on the domain.

 

h. Use y'= 2 / x^1/2 (4-x) ^3/2 to determine the intervals where y is decreasing
----Work this one out please?

From the previous step, the function is never decreasing.

i. Give the coordinates, both x and y, of the relative maximum, if any. (apparently there is no max..
----Work this one out please?
The function is always increasing, to infinity at 4, so there is no maximum.

j. Give the coordinates, both x and y, of the relative minimum, if any. (it's a min)
----Work this one out please?

Since the function is always increasing, the minimum is at the left endpoint.

y(0) = 0 is the minimum

(0, 0)

k. Use y''= 4(x-1)/x^3/2(4-x)^5/2 to determine the open intervals where y is concave up (no concave up since it's all min)
----Work this one out please?

Set your second derivative equal to zero to find your inflection points. In this case, you get x=1 as an inflection point. Your intervals to check inflection on are (0,1) and (1,4)

Plug in values for each interval to determine concavity.

y''(1/2) = 4(1/2 - 1)/(1/2)^3/2(4-1/2)^5/2

Again, you don't need to know the actual number, just its sign. The numerator is negative, and the denominator is positive, so the value is negative. It is concave down on (0,1).

y''(2) = 4(2 - 1) / (2)^3/2 (4-2)^5/2

This is positive, so

y is concave up on (1,4)

l. Use y''= 4(x-1)/x^3/2(4-x)^5/2 to determine the open intervals where y is concave down. (it's concave down)
----Work this one out please?
From the previous section, y is concave down on (0,1)

 

m. Give the coordinates of inflections points, if any. (No inflection points, how do you work this out? )
----Work this one out please?

Again from part k, the inflection point is at x=1. y(1) = √(1 / (4-1)) = √(1/3)

(1, √(1/3))

 

 

It looks like you've got answers to the wrong problem here. The derivatives to the function I have match those quoted in parts h and k. Regardless, I hope this helps you figure out the process, even if there is an issue with the answers you've got compared to what I got. Graphing the equation I worked with on a graphing calculator or online agrees with my answers.