# finding k (a level) (solved)

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#1
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.
Last edited by Navboi; 4 weeks ago
0
4 weeks ago
#2
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.
0
#3
(Original post by sunny.side.up)
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.
Wait I'm very confused, how did you get rid of all the higher powers of x?
0
4 weeks ago
#4
(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.
0
4 weeks ago
#5
0
4 weeks ago
#6
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.
Last edited by MiladA; 4 weeks ago
1
4 weeks ago
#7
(Original post by sunny.side.up)
I'd assume it's a level as it says in the thread title.
0
4 weeks ago
#8
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.
0
4 weeks ago
#9
(Original post by sunny.side.up)
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.
You seem to be ignoring the powers of x here! is not the same as x. This is a quartic function, not a linear one 0
4 weeks ago
#10
(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 0
#11
(Original post by davros)
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 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.
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.
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