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Math Help - Calculus Critical Values, Absolute Min, Max?

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    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.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Luck! View Post
    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?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Luck! View Post
    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 x be the side length of the base.
    Let y be the height

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

    this is our constraint, and from it we can deduce that 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:

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

    \Rightarrow S = 2x^2 + 4xy

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

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

    this is the function we want to minimize, just find its minimum value
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    Quote Originally Posted by Jhevon View Post
    what does this have to do with exponential functions? this is an optimization problem, and no exponential functions are involved

    Let x be the side length of the base.
    Let y be the height

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

    this is our constraint, and from it we can deduce that 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:

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

    \Rightarrow S = 2x^2 + 4xy

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

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

    Let x be the side length of the base.
    Let y be the height

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

    this is our constraint, and from it we can deduce that 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:

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

    \Rightarrow S = 2x^2 + 4xy

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

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

    this is the function we want to minimize, just find its minimum value
    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?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Luck! View Post
    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
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