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Edexcel Math AS (C2) - Differentiation doubt. watch

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    Here are some questions I failed to answer, I'd appreciate if you could help me out.

    Find the values of x for which f(x) is an increasing function, given that f(x) equals:

    a. 3 + 3x - 3x^2 + x^3

    b. 5x^3 + 12x

    c. x^4 +2x^2

    d. x^4 - 8x^3
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    f(x) is an increasing function when f'(x) is non-negative, basically when f(x) is either staying still or has positive gradient
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    (Original post by 13 1 20 8 42)
    f(x) is an increasing function when f'(x) is non-negative, basically when f(x) is either staying still or has positive gradient
    i.e f'(x) is supposed to be greater than zero, right?
    I did that for the rest and it worked out, but for these four, I'm a bit confused.
    If you have time, can you please solve them for me?
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    (Original post by Loody Nagy)
    i.e f'(x) is supposed to be greater than zero, right?
    I did that for the rest and it worked out, but for these four, I'm a bit confused.
    If you have time, can you please solve them for me?
    f'(x) = 0 should be allowable as well, but I can't remember what they want at A level, they might accept either

    What is it about these as compared to the others that is more problematic? The first is just differentiating and then factorising a quadratic, the answer to the second is immediate after differentiating, the third and fourth are also differentiating and factorising; in general look at the factors and consider when each is zero and positive/negative
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    (Original post by 13 1 20 8 42)
    f'(x) = 0 should be allowable as well, but I can't remember what they want at A level, they might accept either

    What is it about these as compared to the others that is more problematic? The first is just differentiating and then factorising a quadratic, the answer to the second is immediate after differentiating, the third and fourth are also differentiating and factorising; in general look at the factors and consider when each is zero and positive/negative
    As for the first sum, I differentiated and factorised.
    I got 3(x-1)^2 > 0
    From this, I concluded that x>1
    However, the answer is: x ∈ ℝ, x ≠ 1 which I don't understand really.

    For the second sum, I don't understand why the answer is immediate after differentiating.
    I got 15x^2 + 12 > 0,
    This is the answer, but I don't get why we are not taking it a step further as in:
    x^2 > -12/15


    For the third sum,
    I got 4x(x^2 +1), from which Ii concluded that x > 0 and x^2 > -1
    However, the answer is:
    x^2 + 1 > 0 for all x and x > 0

    And as for the fourth, I just got it.
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    (Original post by Loody Nagy)
    As for the first sum, I differentiated and factorised.
    I got 3(x-1)^2 > 0
    From this, I concluded that x>1
    However, the answer is: x ∈ ℝ, x ≠ 1 which I don't understand really.

    For the second sum, I don't understand why the answer is immediate after differentiating.
    I got 15x^2 + 12 > 0,
    This is the answer, but I don't get why we are not taking it a step further as in:
    x^2 > -12/15


    For the third sum,
    I got 4x(x^2 +1), from which Ii concluded that x > 0 and x^2 > -1
    However, the answer is:
    x^2 + 1 > 0 for all x and x > 0

    And as for the fourth, I just got it.
    The point is that f'(x) = 3(x - 1)^2 is always non-negative so for all x the function is increasing
    Because x^2 > -12/15 doesn't give us any information about x, as x^2 is more than or equal to 0 anyway for real x. We know immediately that whatever x we pick we'll get a non-negative f'(x) so the function is increasing. Saying x^2 > -12/15 is basically just the same as saying x is any real number in this context
    The third again you are doing unnecessary rearrangement; we are only interested in the value of f'(x), not trying to get things in terms of x or x^2. We know that x^2 + 1 > 0 for all x and that therefore the whole expression is positive whenever x is positive
 
 
 
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