Proving inequalities in PM and SHM questions (1 Viewer)

[~ ReNcH ~]

I've noticed that a few past HSC Q7s have required you to prove an inequality of some sort in either a Projectile Motion or SHM question e.g.
Prove x(x-y²) > xy.
Do these generally require the use of the discriminant to prove? Are there many other ways in which an inequality could arise?

 

[cj_bridle]

are there any examples you can think of? i havent come accross any at all.

 

[mojako]

>> Are there many other ways in which an inequality could arise? <<
We can make infinetely many inequalities.

>> Do these generally require the use of the discriminant to prove? <<
Not really.
One of the common methods is to complete the square and show LHS - RHS > 0 or something along those lines.

Another technique is to examine the graphs of a curve and compare the exact area under the curve with an approximate area, such as in Q6 (b) 2002 HSC. But this thing normally comes in isolation, not in questions involving motion etc.

 

[Archman]

one fact you can use is to look at an equation as an quadratic in something, so the discriminant >= 0

 

[mojako]

actually i've never used the discriminant technique
might be useful in 4U... hope so.

anyway, taking discriminant for x(x-y2) > xy is a bit of a subtle point.. I'll try to explain it in case anyone didnt know already

let us review what we usually do with discriminant.
suppose y = ax^2 + bx + c
if the discriminant is <0 then the curve never touches the x-axis.
alternatively, the value of "ax^2 + bx + c" never equals zero.

For x(x-y2) > xy,
LHS - RHS
= x^2 - x*y^2 - xy (after rearranging)
= x^2 - (y^2+y)*x (making it a quadratic in x... you can also make it in y)

if the discriminant is <0, the value of "x^2 - (y^2+y)*x" never equals zero.
Actually.. I think it is either always negative or always positive.
So you need to test for a random x and y pair, and if it's positive it will be positive for any x and y, whilst if it's negative it will be negative for any x and y.

 

[Estel]

You don't need to test if it is always positive or negative.
Rather, just state that it's positive definite

 

[mojako]

You don't need to test if it is always positive or negative.
Rather, just state that it's positive definite

hmm.. why?

I just realised.. for Rench's Q the discriminant is positive...
and by inspection,
or alternatively by putting x=1 and y=2,
the statement x(x-y^2) > xy doesn't hold.

 

[Estel]

Think about it:
if a polynomial has no real roots and a positive leading coefficient, how can it ever be negative?

 

[mojako]

Think about it:
if a polynomial has no real roots and a positive leading coefficient, how can it ever be negative?

at first I thought yes.
but,
what about:
writing x^2 - (y^2+y)*x as a quadratic in y?
you get -x as the leading term, which can be negative and positive
this is confusing

 

[Archman]

what are you trying to prove again?

 

[Estel]

Archman: the result was untrue anyway so the original q is rather irrelevant

Mojako: the example you gave doesn't necessarily have a negative discriminant, nor does it necessarily have a positive leading coefficient and hence it is irrelevant
[Seriously, I think I've misinterpreted what you've typed... please elaborate]

 

[mojako]

the example you gave doesn't necessarily have a negative discriminant

it's Rench's example.
I just used it to show how we can use discriminant and the logic behind it.
but it happens that the discriminant in that example is in fact not negative.

when I said:

what about:
writing x^2 - (y^2+y)*x as a quadratic in y?
you get -x as the leading term, which can be negative and positive

I tried to point out the fact that a positive leading term of a quadratic in x doesn't necessarily mean a positive leading term of the equivalent quadratic in y. But I'm not sure about this myself.

 

[nit]

Basically every one of our last couple of school 3 unit maths exams has had a projectile motion inequality involving use of the discriminant. So, as far as we're concerned, it's rather useful...

 

[ngai]

Prove x(x-y²) > xy.

even the simple x=y=0 blows up this question

 

[Archman]

even the simpler x = 0 does.

 

[CrashOveride]

I've done the last 10 years and there was only one kind of question o this type anyway. And i dont see how using discriminant would have helped...all u had to do was show some stuff > other stuff hence other stuff > more stuff and other random...related...things.

Also, would i be fair to say in 3unit and even 4unit there seems to be ...styles emerging depending on the year blocks? Say the last 3 years in 3unit i think were easier than 99-96 or something of that nature....the earlier 3unit exams i think ..not all but mostly are harder than most recents ones... discuss.

 

[~ ReNcH ~]

it's Rench's example.
I just used it to show how we can use discriminant and the logic behind it.
but it happens that the discriminant in that example is in fact not negative.

when I said:

I tried to point out the fact that a positive leading term of a quadratic in x doesn't necessarily mean a positive leading term of the equivalent quadratic in y. But I'm not sure about this myself.

I actually just made that up on the spot (not a real example from a book or anything). Whether it can be solved or not, I wouldn't have a clue since it was just a random inequality that I made up. But it seems to make for an interesting question