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Math Help - Using first/second derivates

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    Using first/second derivates

    This problem has to do with the second derivative of a function. Here's the problem:



    A function f has f(3) = 20, f\prime(3) = 2, and f\prime\prime(x) > 0 for x > 3. Which of the following are possible values for f(5)? State why or why not.

    a.) 26


    b.) 24


    c.) 22





    (Note: It's NOT multiple choice. It's asking you to consider each on of these values, say whether or not they would be possible values for f(5), and then explain why you said yes or no. Just to clarify...)



    What I've Done So Far: I attempted to graph f(x), because I know that the point (3 , 20) is on it, and the tangent line to that point has a slope of 2. I also know the graph is increasing, and is concave up from (3 , \infty), implying that before that point, it's concave down (...right?). Having graphed it...I'm lost. I have no idea how to determine whether or not these are possible values. My best guess was to base it off of the slope (being 2). According to my poorly drawn graph, it doesn't look like b or c would be possible because of how the graph is increasing. But it seems incorrect to just eyeball it like that.

    Any help would be greatly appreciated!
    Last edited by lysserloo; September 30th 2009 at 06:15 PM.
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    Quote Originally Posted by lysserloo View Post
    This problem has to do with the second derivative of a function. Here's the problem:



    A function f has f(3) = 20, f\prime(3) = 3, and f\prime\prime(x) > 0 for x > 3. Which of the following are possible values for f(5)? State why or why not.

    a.) 26


    b.) 24


    c.) 22






    What I've Done So Far: I attempted to graph f(x), because I know that the point (3 , 20) is on it, and the tangent line to that point has a slope of 2. I also know the graph is increasing, and is concave up from (3 , \infty), implying that before that point, it's concave down (...right?). Having graphed it...I'm lost. I have no idea how to determine whether or not these are possible values. My best guess was to base it off of the slope (being 2). According to my poorly drawn graph, it doesn't look like b or c would be possible because of how the graph is increasing. But it seems incorrect to just eyeball it like that.

    Any help would be greatly appreciated!
    if f(x) was linear, f(5) would equal 26 ... since f'' > 0 for all x > 3, f is increasing at an increasing rate ... f(5) would actually be greater than 26.
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    Quote Originally Posted by skeeter View Post
    if f(x) was linear, f(5) would equal 26 ... since f'' > 0 for all x > 3, f is increasing at an increasing rate ... f(5) would actually be greater than 26.

    Okay...so then

    a.) does that mean none of these values are possible?

    b.) how do you figure that if f(x) is linear it would be 26? Is it because the slope at x = 3 is 2, so you're just going up 2 over 1 for x = 4 then x = 5?


    Thank you for your reply! ....I'm still confused as to how I should answer this question though.
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    Quote Originally Posted by lysserloo View Post
    Okay...so then

    a.) does that mean none of these values are possible?

    b.) how do you figure that if f(x) is linear it would be 26? Is it because the slope at x = 3 is 2, so you're just going up 2 over 1 for x = 4 then x = 5?

    correction ... up 3 and over 1, because f'(3) = 3


    Thank you for your reply! ....I'm still confused as to how I should answer this question though.
    bad multiple choice question, imho ... 26 would be the "best" answer if those are the rules of the game.

    this isn't one of those "online" question, is it?
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    Quote Originally Posted by skeeter View Post
    bad multiple choice question, imho ... 26 would be the "best" answer if those are the rules of the game.

    this isn't one of those "online" question, is it?
    Oh gosh, in regard to your correction, no, it's f\prime(3) = 2. I wrote the wrong thing on the original problem. Crap! That probably changes things.

    And no, it's not a web problem. Just a particularly confusing one on a homework assignment I have. It's not multiple choice either. It's saying that you have to consider each of those values, and say whether or not they would be possible, and why.
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    Quote Originally Posted by lysserloo View Post
    Oh gosh, in regard to your correction, no, it's f\prime(3) = 2. I wrote the wrong thing on the original problem. Crap! That probably changes things.

    And no, it's not a web problem. Just a particularly confusing one on a homework assignment I have. It's not multiple choice either. It's saying that you have to consider each of those values, and say whether or not they would be possible, and why.
    if that is the case, f'(3) = 2 , then f(5) > 24 using my previous logic.
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    Quote Originally Posted by skeeter View Post
    if that is the case, f'(3) = 2 , then f(5) > 24 using my previous logic.

    Hmm. Okay then I have one last question, if you don't mind! (You've already been so helpful.)

    How exactly would I explain why it's NOT 22 or 26 using the information given?
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    Quote Originally Posted by lysserloo View Post
    Hmm. Okay then I have one last question, if you don't mind! (You've already been so helpful.)

    How exactly would I explain why it's NOT 22 or 26 ??? using the information given?
    f(5) could be 26 ... note that I said f(5) is greater than 24, because if f(x) were linear (that would be increasing at a constant rate), then f(5) = 24 , but since f''(x) > 0 for all x > 3, f is increasing at an increasing rate.
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    Quote Originally Posted by skeeter View Post
    f(5) could be 26 ... note that I said f(5) is greater than 24, because if f(x) were linear (that would be increasing at a constant rate), then f(5) = 24 , but since f''(x) > 0 for all x > 3, f is increasing at an increasing rate.
    OH! Okay yeah, sorry. Math is just one of those things that never "clicks" with me.

    I know what I'm doing now.

    Thank you SO much. I appreciate your patience.
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