4U Maths Question Thread

[maria.micale]

Hi, there is this inequality question I am having trouble with. Could someone please assist?

Using a^2 + b^2 + c^2 >or= ab + bc + ca,
Show that a^2 + b^2 + c^2 >or= 3*cube root(a^2*b^2*c^2)

[mxrylyn]

Hello

I am having trouble with seeing how (-2 plus/minus root 8 x i) over 2,
can be written as -1 plus/minus root 2 x i

[RuiAce]

Hello

I am having trouble with seeing how (-2 plus/minus root 8 x i) over 2,
can be written as -1 plus/minus root 2 x i

[RuiAce]

Hi, there is this inequality question I am having trouble with. Could someone please assist?

Using a^2 + b^2 + c^2 >or= ab + bc + ca,
Show that a^2 + b^2 + c^2 >or= 3*cube root(a^2*b^2*c^2)

What is the source of this question? I know I'm quite braindead but I genuinely don't see how the first inequality is supposed to be of any use right now

(More or less because you're basically trying to prove the AM-GM inequality for three variables. But \(a^2+b^2+c^2 \ge 3\sqrt[3]{a^2b^2c^2} \) and \(ab+ac+bc \ge 3\sqrt[3]{a^2b^2c^2}\) are both direct applications of the AM-GM inequality anyway.)

[maria.micale]

Show that if a, b, c, d > 0, then:
a/b + b/c + c/a + d/a  >or= 4

Just started doing advanced 3 unit inequalities and I have no idea what to do with this question, any help would be appreciated,
cheers.

The question I asked was from Fitzpatrick. The question above is also from this textbook. Do you know how to solve either of them. If so could you please show working because I am stumped.

Thanks

[RuiAce]

Oh wow, I completely missed that there was another question on the previous page. I'll address your's soon.

Although, having said that,

Show that if a, b, c, d > 0, then:
a/b + b/c + c/a + d/a  >or= 4

Just started doing advanced 3 unit inequalities and I have no idea what to do with this question, any help would be appreciated,
cheers.

This is actually one of the questions covered in my 4U notes book. So all I will do is put the working out here.



___________________________________

Gonna just edit this post because there's no point making another one when I don't have a concrete answer.

The question I asked was from Fitzpatrick. The question above is also from this textbook. Do you know how to solve either of them. If so could you please show working because I am stumped.

Thanks

With that question you asked, the only logical thing to do is to somehow prove that \( ab+ac+bc\ge 3\sqrt[3]{a^2b^2c^2} \), because otherwise you aren't using the hence method. But to do that, you would have to make a very weird substitution to force certain terms to collapse (in your given inequality), and right now I can't see how that works. I won't say that the question is nonsensical yet, but it seems extremely peculiar. (They actually make you prove something using a HORRIBLE method.)

[RuiAce]

Remark: In the HSC, you would be guided through it. This method is quite famous but you're not expected to know it off by heart.

[MLov]

Recall that for a rectangular prism, given a fixed volume, the sum of dimensions is minimised when it is a cube.
Consider a rectangular prism with sides a²,b² and c² and a cube with sides . It is not hard to check they have the same volume.
Therefore for any a,b,c: as required.

[RuiAce]

Recall that for a rectangular prism, given a fixed volume, the sum of dimensions is minimised when it is a cube.
Consider a rectangular prism with sides a²,b² and c² and a cube with sides . It is not hard to check they have the same volume.
Therefore for any a,b,c: as required.

This is really just bringing back memories of Lagrange multipliers...

[itssona]

Mod edit: Was authorised to fix up the LaTeX

[RuiAce]

Mod edit: Was authorised to fix up the LaTeX

Personally, however, I hate doing division of coordinates because it can get confusing. (In a way, I need to think of things upside down.) So I'll do this instead.

________________________________________________


So on GeoGebra, you can consider \( f(x) = x^2+x-2 \) and \( g(x) = 1/f(x) \).

On GeoGebra, you can consider \( h(x) = 2x^2-x-3 \)

And then just multiply ordinates.

Remark: Again, division is probably recommended, because it's faster. I just prefer the less brain-work involved with this longer approach

[maria.micale]

Hi, is there a way to find the zeros of polynomials with complex coefficients that have a degree higher than 2? Because I am having trouble solving theses types of polynomials. I'm not sure really where to start with them. Thanks!

[RuiAce]

Hi, is there a way to find the zeros of polynomials with complex coefficients that have a degree higher than 2? Because I am having trouble solving theses types of polynomials. I'm not sure really where to start with them. Thanks!

There is no concrete rule of thumb. It always depends on a question-by-question basis.

[maria.micale]

There is no concrete rule of thumb. It always depends on a question-by-question basis.

But like its not like with polynomials with real coefficients wear you can check if factors of the constant term are zeros right? Like there is nothing like that right?

I have this question,

Find the real numbers k such that z=ki is a root of the equation z^3 + (2+i)Z^2 + (2+2i)z +4 = 0. Hence or otherwise, find the three roots of the equation.

I subbed in ki where there was a z and ended up getting -ik^3 + (-2-i)k^2 + (2i-2)k +4 =0 and I am not sure how to find k from here.

[RuiAce]

But like its not like with polynomials with real coefficients wear you can check if factors of the constant term are zeros right? Like there is nothing like that right?

I have this question,

Find the real numbers k such that z=ki is a root of the equation z^3 + (2+i)Z^2 + (2+2i)z +4 = 0. Hence or otherwise, find the three roots of the equation.

I subbed in ki where there was a z and ended up getting -ik^3 + (-2-i)k^2 + (2i-2)k +4 =0 and I am not sure how to find k from here.

Hint: The fact that \(k\) is real means that you can do some rearranging, and then equate the real and imaginary parts. The real parts should give you a quadratic...