#extreme values, critical numbers, concaves, related rates

78 messages · Page 1 of 1 (latest)

feral inlet
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need help in some of these and (mostly first one) and to check answers in rest

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elfin spadeBOT
hallow totem
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Its optimization, they have pretty much the same methods of solving, it just varies if maximum or minimum is being asked. You can look up on yt for optimization problem and try

feral inlet
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what about the first one

sterile thorn
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Do you know what the slope of an extreme value is?

quasi totem
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solve for critical numbers by finding the first derivative then setting it equal to 0

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analyze concavity inflectivity yada yada by finding the second derivative then factoring

cerulean python
# feral inlet need help in some of these and (mostly first one) and to check answers in rest

does the volune have to do with anything on question 1? if not, use AM-GM with 3 numbers (a^3+b^3+c^3 >= 3abc for a;b;c >= 0 with equality happening at a=b=c or a+b+c = 0 (which can't happen in this situation), try proving that first, then try to think of how you would use it to get rid of the x)

equality happen at x^3 = 50 or x = cuberoot(50) which is in the [1;10] range

sterile thorn
cerulean python
sterile thorn
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don't just give away the answer, guide the OP

cerulean python
sterile thorn
cerulean python
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still don't know why the volume is in there tho maybe it's a red herring

sterile thorn
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square base and a volume of 50 ft³

cerulean python
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i'm thinking of it as a rectangle box so the height could be anything

sterile thorn
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i think so too

cerulean python
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so the 2 sides on top and bottom are square and the 4 other sides are rectangle?

sterile thorn
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otherwise the problem would be very easy

cerulean python
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i'll try that ig

sterile thorn
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as then length = width = height

sterile thorn
cerulean python
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if that was the case then x = 0 which won't make any sense, so i'm really just assuming the volume is a red herring

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oh im assuming S(x) = 2x^2 + 200/x where x is in term of ft,it didn't mention in the question but since the volume is measured in ft x should also be in ft i think

sterile thorn
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the question asks for the minimum surface area on the interval [1, 10]

cerulean python
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i'm just wondering why they list the volume in the first place

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is the surface area the area of 6 sides then?

sterile thorn
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i think to verify your answer

sterile thorn
cerulean python
sterile thorn
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remember, don't just solve it

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explain what steps to take

cerulean python
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i'll just post the answer in spoiler

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and give hint in non-spoiler

sterile thorn
bold oasis
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because thats where the curve is level

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meaning its the bottom or top point of the function

sterile thorn
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so that's where the extreme values are

cerulean python
# cerulean python does the volune have to do with anything on question 1? if not, use AM-GM with 3...

i'm pretty sure this way to solve is actually correct atleast to my interpertation of the question, here's my solution

|| let's call the side of the square x (ft), and the height h (ft)

volume = 50 ft^3, therefore x^2h = 50 ft^3
surface area = 2x^2 + 4xh

S(x) is the surface area, therefore

2x^2 + 4xh = 2x^3 + 200/x
and x is not 0, so 2x^3 + 4x^2h = 2x^3 + 200, plug x^2h = 50ft^3 we have
2x^3 + 200 = 2x^3 + 200 which is always true, so we just have to calculate the minimum of 2x^2 + 200/x and find the value of x where equality happen then solve for h

with a;b;c >= 0, a^3+b^3+c^3 - 3abc = 1/2 (a+b+c)*((a-b)^2+(b-c)^2+(c-a)^2) >= 0 therefore a^3 + b^3 + c^3 >= 3abc or a +b + c >= 3cuberoot(abc)
apply to the question we have, 2x^2 + 200/x = 2x^2 + 100/x + 100/x >= 3cuberoot(20000)

equality happens when 2x^2 = 100/x or x^3 = 50
x = cuberoot(50) ft which is in the [1;10] range so it is valid
h = cuberoot(50) ft ||

sterile thorn
cerulean python
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my interpertation is that the box is this shape

image.png
sterile thorn
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that's my interpretation as well

sterile thorn
cerulean python
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well yeah what i did also get the minimum

sterile thorn
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as the other questions are also about extreme values

cerulean python
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well here's the answer i get i guess, you guys can check if it's the same

Absolute minimum || 3cuberoot(20000) ||
Dimension where absolute minimum happens || The box is a cube with a side length of cuberoot(50) ||

sterile thorn
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you still need to simplify

cerulean python
sterile thorn
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the cube roots

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pretty sure the first can be simplified

cerulean python
# sterile thorn the cube roots

okay then

Absolute minimum ||30*cuberoot(20)||
Dimension where minimum happens ||The box is a cupe with a side length of cuberoot(50)||

sterile thorn
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according to my calculator, your solution doesn't

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it's too big

cerulean python
sterile thorn
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good point, now i'm not sure

cerulean python
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idk how it can be from 0 to 10 though, the value i calculated is already the lowest so im not sure how to go lower

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maybe it can with negatives?

sterile thorn
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how can a length or volume be negative

cerulean python
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the minimum value of 2x^2 + 200/x is already way larger than 10 for all x > 0 and there's also more conditions on top of that (tbf in my solution the volume doesn't affect anything as you can still find a valid value of the height where it works)

sterile thorn
sterile thorn
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but the solutions are in ft^2 and ft

sterile thorn
cerulean python
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Alright then

cerulean python
sterile thorn
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because it's the formula for the surface area

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but you can subtract the base area term from both sides

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i think

cerulean python
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Yeah but I need to prove S(x) is always gonna be equal to the surface area first so I can make sure that it's possible to find h once I find the equality for min S(x)

sterile thorn
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but the question states that it is the formula for the surface area

cerulean python
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If it's smt like S(x) = x^2 + 250/x it'll only be true for a certain value of x