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Edexcel Math AS (C2) - Differentiation doubt.

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
Reply 1
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math42
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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
Reply 2
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Loody Nagy
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Original post by 1 8 13 20 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?
Reply 3
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math42
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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
Reply 4
Avatar for Loody Nagy
Loody Nagy
OP
3
Original post by 1 8 13 20 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.
Reply 5
Avatar for math42
math42
20
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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