# Thread: Calculus Critical Values, Absolute Min, Max?

1. ## Calculus Critical Values, Absolute Min, Max?

I am having some problems with some homework questions, I was hoping maybe somebody could help me work these out so I had some reference material for the rest of my homework.

Find the absolute maximum, and minimum, if any, on the interval [0,9]
f(x)=1/8x^2 - 4sqrtx

need f prime of (x)
critical values
absolute min, and max

and

Exponential Functions

If an open box has a square base and a volume of 108 inches cubed, and is constructed from a sheet of tin, find the dimensions of the box that will use the minimum amount of material in it's construction.

2. Originally Posted by Luck!
I am having some problems with some homework questions, I was hoping maybe somebody could help me work these out so I had some reference material for the rest of my homework.

Find the absolute maximum, and minimum, if any, on the interval [0,9]
f(x)=1/8x^2 - 4sqrtx

need f prime of (x)
critical values
absolute min, and max
step 1: find the corresponding y-values for the end points, that is, find f(0) and f(9)

step 2: find f'(x), set it equal to zero and solve for x. these are the critical values

step 3: find the corresponding y-value for EACH critical x-value.

step 4: compare all the y-values you have found so far. the largest one is the absolute max, the smallest one is the absolute min.

can you continue?

3. Originally Posted by Luck!
Exponential Functions

If an open box has a square base and a volume of 108 inches cubed, and is constructed from a sheet of tin, find the dimensions of the box that will use the minimum amount of material in it's construction.
what does this have to do with exponential functions? this is an optimization problem, and no exponential functions are involved

Let $\displaystyle x$ be the side length of the base.
Let $\displaystyle y$ be the height

Then we have the volume given by: $\displaystyle V = x^2 y = 108$

this is our constraint, and from it we can deduce that $\displaystyle y = \frac {108}{x^2}$

now we want to construct a (i'm assuming closed top) box with the above volume and minimal surface area.

the surface area of this box is given by:

$\displaystyle S = \underbrace {x^2 + x^2}_{\mbox{area of base and top}} + \underbrace {xy + xy + xy + xy}_{\mbox{area of the other 4 sides}}$

$\displaystyle \Rightarrow S = 2x^2 + 4xy$

since $\displaystyle y = \frac {108}{x^2}$, we have:

$\displaystyle S = 2x^2 + \frac {432}x$

this is the function we want to minimize, just find its minimum value

4. Originally Posted by Jhevon
what does this have to do with exponential functions? this is an optimization problem, and no exponential functions are involved

Let $\displaystyle x$ be the side length of the base.
Let $\displaystyle y$ be the height

Then we have the volume given by: $\displaystyle V = x^2 y = 108$

this is our constraint, and from it we can deduce that $\displaystyle y = \frac {108}{x^2}$

now we want to construct a (i'm assuming closed top) box with the above volume and minimal surface area.

the surface area of this box is given by:

$\displaystyle S = \underbrace {x^2 + x^2}_{\mbox{area of base and top}} + \underbrace {xy + xy + xy + xy}_{\mbox{area of the other 4 sides}}$

$\displaystyle \Rightarrow S = 2x^2 + 4xy$

since $\displaystyle y = \frac {108}{x^2}$, we have:

$\displaystyle S = 2x^2 + \frac {432}x$

this is the function we want to minimize, just find its minimum value

No idea why its called that, I took it straight out of my math book. Unfortunately my math book doesn't have any examples..minus the simplest forms..which I understand to a degree. It is just that when I actually have to more difficult problems..it stinks..I have no reference material.

5. Originally Posted by Jhevon
what does this have to do with exponential functions? this is an optimization problem, and no exponential functions are involved

Let $\displaystyle x$ be the side length of the base.
Let $\displaystyle y$ be the height

Then we have the volume given by: $\displaystyle V = x^2 y = 108$

this is our constraint, and from it we can deduce that $\displaystyle y = \frac {108}{x^2}$

now we want to construct a (i'm assuming closed top) box with the above volume and minimal surface area.

the surface area of this box is given by:

$\displaystyle S = \underbrace {x^2 + x^2}_{\mbox{area of base and top}} + \underbrace {xy + xy + xy + xy}_{\mbox{area of the other 4 sides}}$

$\displaystyle \Rightarrow S = 2x^2 + 4xy$

since $\displaystyle y = \frac {108}{x^2}$, we have:

$\displaystyle S = 2x^2 + \frac {432}x$

this is the function we want to minimize, just find its minimum value
$\displaystyle S = 2x^2 + \frac {432}x$
How on earth do you find the minimum? I understand that this is the "function" of figuring out the minimum amount of the cube but what do you plug in?

6. Originally Posted by Luck!
$\displaystyle S = 2x^2 + \frac {432}x$
How on earth do you find the minimum? I understand that this is the "function" of figuring out the minimum amount of the cube but what do you plug in?
how do you find the local minimum of any function in calculus (if it exists)? find the derivative and set it equal to zero. take the answer that makes sense