Given that the distance AB = 4root2, find k

The question: https://media.discordapp.net/attachments/480833140991066123/560615165838491658/20190328_000415.jpg?width=383&height=514

I got the first part right, that being finding the second derivative of the equation to get to -4-6x = 0 and rearranging to get x = -2/3

However, the next question confused me for a while and one of the ways I tried to answer this question was this:

Integrating original gives f(x) = kx -0.5x^2-x^3
x(k-0.5x-x^2) = 0
Use the discriminant to find k, end up getting -0.25 = 4k and k = -1/8 which is just wrong.

Reply 1

begbie68

so you have an expression for f'(x) and you need to find the roots of f(x), other than x=0

let's call them x1 and x2
then we know that |x1 - x2| = 4root2

we also know the result of x(x-x1)(x-x2) in a different form (or at least we can work it out)

So, you should know what to do now. NB Be careful with your +/- signs.

This is just one method. There is a more elegant method of solution that you may decide to consider.

Reply 2

RDKGames

Study Forum Helper

Original post by OJ Emporium

Do you understand what the discriminant even represents? It's not appropriate here.

The quadratic

k - 0.5x - x^2

needs to have two roots at A and B. When you think of a quadratic, you should realise that the x-coordinate that is the midpoint of the two roots is precisely where the max/min of the quadratic occurs, and if you draw a vertical line through this point you would split the quadratic curve into two perfect halves.

So, via quick differentiation, you can determine the x-coordinate where the quadratic has its turning point. Then the two roots (due to symmetry) must be at an equal distance of

\dfrac{1}{2}(4\sqrt{2})

on either side of this x-coordinate... hence giving you them roots.

Then the product of these roots is precisely

k

Reply 3

begbie68

RDK : we CAN use the discriminant (which involves k) along with the 4root2 (given) to find k.

However, it's not the 'usual' application of discriminant which most students just identify with the number of solutions to the quad.

We COULD define discriminant in terms of the distance btwn the roots of a quad (the precise relationship I'll leave to the OP to determine)

in that case, if

discriminant > 0, then there is a positive distance btween the roots <=> 2 real roots

discriminant = 0, then there is NO dist. btwn the roots <=> one real root (rptd solution)

discriminant < 0, then a negative dist btwn roots, which is not 'valid' <=> no real roots

Reply 4

OJ Emporium

Not gonna lie im still confused lol

Reply 5

begbie68

maybe because you integrated wrongly ... have a look at your x-squared term ...

Original post by OJ Emporium

Not gonna lie im still confused lol

Reply 6

OJ Emporium

Original post by begbie68

maybe because you integrated wrongly ... have a look at your x-squared term ...

Yikes i noticed that, the correct integral is kx - 2x² -x³

Reply 7

OJ Emporium

Nvm, I realised I could use the quadratic formula to just find both the positive and negative x values and subtract them from each other to find k, being 7. I'm not sure whether this is correct though

Reply 8

begbie68

yes, it is correct.

You could also try (x-A)(x-B) = k - 2x - x^2 and solve for A,B in terms of k

eg, is it (x+A)(x-B)? or do we change signs?

Most students I've seen try this method (for this question) make a mess of the signs. Hence why I said be careful with your signs.

Original post by OJ Emporium