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finding k (a level) (solved)

Hi,

I'm not sure on where I would start with the following question:

I would like some help on where I should start first and why I should do those steps

Any help is appreciated, thank you.

(edited 3 years ago)

Reply 1

username5347244

First, you should add the variables together. So, 3x + (-4x) + (-12x). Two negatives make a positive.
Then, you would make your new equation. So, k = 19x + 20
Next, you have to solve.

k = 19x + 20
-19 -19
19x = 1
/19 /19

k = 0.053

So, the answer would be A.

Reply 2

Navboi

Original post by sunny.side.up

Wait I'm very confused, how did you get rid of all the higher powers of x?

Reply 3

username5347244

Original post by Navboi

Wait I'm very confused, how did you get rid of all the higher powers of x?

Because you have to add them all together, so it turns into only 1 x with all of the added variables.

Reply 4

username5347244

May I ask what grade your in?

Reply 5

MiladA

A quartic polynomial with four distinct real roots has a similar shape to the letter 'W'. For there to be exactly four distinct real roots the line y=k needs to intersect the curve four times horizontally.

This occurs when the line is below the middle peak of the letter W and above the two bottom peaks. These peaks are locations of max/min points of the curve respectively. Therefore, establish the derivative of the function 3x^4-4x^3-12x^2+20-k and set equal to zero to obtain the x values for which this holds.

Determine the value of the function at each x coordinate and then establish the bounds for which there are four distinct real roots. It's helpful to sketch the curve.

(edited 3 years ago)

Reply 6

MiladA

Original post by sunny.side.up

May I ask what grade your in?

I'd assume it's a level as it says in the thread title.

Reply 7

username5347244

Original post by MiladA

I'd assume it's a level as it says in the thread title.

Oh, I didn't realize. It didn't say that before.

Reply 8

davros

Original post by sunny.side.up

You seem to be ignoring the powers of x here!

x^4

is not the same as x. This is a quartic function, not a linear one :smile:

Reply 9

davros

Original post by Navboi

Hi,

I'm not sure on where I would start with the following question:

I would like some help on where I should start first and why I should do those steps

Any help is appreciated, thank you.

As suggested earlier, drawing a rough sketch of a quartic function is going to help you a lot here. Remember that x^4 is large and positive when x is large and positive, and also when x is large and negative. So your function is probably going to be an approximate 'W' shape (although it could be a 'U' or something slightly different if there were multiple roots).

Differentiate first of all to find the turning points. You will find that you can take out some obvious factors, and what's left has nice roots. When you plug in the x-coordinates of the turning points into the original function to find the corresponding y-values, you'll see what the question is asking :smile:

Reply 10

Navboi

Original post by davros

Original post by MiladA

Thank you guys! I tried to differentiate it at first but as this was a non-calculator uni admissions paper, I got intimidated when I saw the cubic and instantly assumed I was doing something wrong, not realising it could be easily factorised.