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Math Help - problem of increasing function

  1. #1
    Senior Member Sambit's Avatar
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    Question problem of increasing function

    Find the range of the values for c for which f(x) = x^3 - 3x^2 + 3cx + 1 is strictly increasing.
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  2. #2
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    Find f'(x). For f(x) to be strictly increasing, f'(x) > 0 for all x.
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  3. #3
    Senior Member Sambit's Avatar
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    yes i know that. but writing f'(x) > 0 leads to:-
    3x^2 - 6x + 3c > 0. after this how can i solve this for  c ?
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  4. #4
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    Use the quadratic formula, pay attention in particular to the discriminant.

    Think about what it means to have f'(x) > 0. It means if you graph f'(x), you should have every single point above the x axis => no real roots.

    So under what conditions does a quadratic equation not have real roots?
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  5. #5
    Senior Member Sambit's Avatar
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    a quadratic equation does not have real roots when the discriminant is < zero. so, here,  4 < 4c, ie,  c>1. am i correct?
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  6. #6
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    yes
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  7. #7
    Senior Member Sambit's Avatar
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    but tell me one thing....here we are considering the non-existence of any real root of the quadratic equation because we have a > 0 condition. but in case we had the function decreasing, then also we would have considered the same criterion, that is, the non-existence of any real root; and hence the final answer (ie, c > 1) would be the same.

    how can you explain this?
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  8. #8
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    If we had a decreasing function (to be more precise, polynomials), you cannot have a positive leading coefficient for for f'(x). Simply because the limit as x gets very large will be positive infinity.
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  9. #9
    Senior Member Sambit's Avatar
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    ok. so the fact is the original equation, ie,  f(x) = x^3 - 3x^2 + 3cx + 1 can never be MONOTONICALLY DECREASING, but may be decreasing in some interval. is it?

    ok. i am waiting for your reply
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  10. #10
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    Quote Originally Posted by Sambit View Post
    ok. so the fact is the original equation, ie,  f(x) = x^3 - 3x^2 + 3cx + 1 can never be MONOTONICALLY DECREASING, but may be decreasing in some interval. is it?
    Yes it will decrease at some interval, but never monotonic decreasing.
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  11. #11
    Senior Member Sambit's Avatar
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    ok.. thanks a lot for spending so much time for my thread ...
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  12. #12
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    Don't worry about it. Just another procrastination technique I have.
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