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Math Help - Functions and derivatives. Any experts out there?

  1. #1
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    Talking Functions and derivatives. Any experts out there?

    I'm confused with this whole concept of functions and derivatives. If you could help me that'd be wonderful. Thank you so much!

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    Suppose that f and f ' are defined for all values of x...

    f' has the following properties:
    f' (x) > 0 for all x
    f' has a maximum value of 5. This value occurs at the approximate x-coordinate 0.347
    As x increases towards infinity, f ' (x) decreases monotonically toward 0.
    As x decreases towards negative infinity, f ' (x) decreases monotonically toward 0.
    Also, f (x) =2.5

    1) explain why f (x) could not be the function ax^5 + bx^4 + cx^3 + dx^2 + ex + f for any values of A, B, C, D, E, and F.
    2) Use the "speed limit principle" to show that the value of f(1) must be less than 6.
    3) Use the "speed limit principle" to find upper and lower bounds on the value of f(0).
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    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by lilxswttooth View Post
    I'm confused with this whole concept of functions and derivatives. If you could help me that'd be wonderful. Thank you so much!

    ------

    Suppose that f and f ' are defined for all values of x...

    f' has the following properties:
    f' (x) > 0 for all x (*)
    f' has a maximum value of 5. This value occurs at the approximate x-coordinate 0.347
    As x increases towards infinity, f ' (x) decreases monotonically toward 0.
    As x decreases towards negative infinity, f ' (x) decreases monotonically toward 0. (**)
    Also, f (x) =2.5

    1) explain why f (x) could not be the function ax^5 + bx^4 + cx^3 + dx^2 + ex + f for any values of A, B, C, D, E, and F.
    2) Use the "speed limit principle" to show that the value of f(1) must be less than 6.
    3) Use the "speed limit principle" to find upper and lower bounds on the value of f(0).
    1) Let f(x):=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f for x\in\mathbb{R}. Clearly, a polynomial (unless identically 0) tends to \pm\infty at \pm\infty. Since f^{\prime} (which is also a polynomial) tends to 0 at -\infty by (**), we must have f^{\prime}\equiv0, and contradicts to (*).
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