The graphs of the functions f(x) = x3 + (a + b)x2 + 3x − 4

[mchallenge77]

The graphs of the functions f(x) = x3 + (a + b)x2 + 3x − 4 and g(x) = (x − 3)2 + 1 touch. Express a in terms of b.

Can I get help with this question, please?

 

[Dr.Peterson]

What help do you need?

Please show us whatever work you have tried, and tell us where you are stuck. It will also help if you tell us the context of the problem -- what topics have you studied? And how do you interpret the statement that the graphs "touch"?

 

[mchallenge77]

Currently doing functions, graphing, grade 10 level, at the moment, it's the part of the cubic functions sub-topic.

I don't understand the statement TOUCH either - does it mean touch once, or twice?

I tried doing it simultaneously.

When solving simultaneously, I got stuck at:

x^3 + (a+b)x^2 + 9x + 6 = 0

I am unsure of what to do after.


They are asking:

The graphs of the functions f(x) = x^3 + (a + b)^2 + 3x − 4 and g(x) = (x − 3)^2 + 1 touch. Express a in terms of b.



the solution at the back of the book says a =(65 – 8h)/8


How should I be looking at this question??

 

[Dr.Peterson]

I would take "touch" to mean that at some point they are tangent to one another in some one place, possibly that being the only point of intersection. Have you dealt with that concept, either in terms of the derivative, or in some other way? What particular things have you learned about cubics?

Playing with the problem on Desmos, I can make a graph that apparently represents the solution:



I graphed f(x) = x^3 + cx^2 + 3x − 4 and found that c = 65/8 (approximately), which agrees with the stated solution.

So that seems to be what the problem means: one point of tangency, but not forbidding another intersection. You'll have to tell me a little more about what tools you have available for handling tangent polynomials. One approach would be to find c such that your equation has exactly two solutions. Do you have a technique for doing that?

 

[Steven G]

Currently doing functions, graphing, grade 10 level, at the moment, it's the part of the cubic functions sub-topic.

I don't understand the statement TOUCH either - does it mean touch once, or twice?

I tried doing it simultaneously.

When solving simultaneously, I got stuck at:

x^3 + (a+b)x^2 + 9x + 6 = 0

I am unsure of what to do after.


They are asking:

The graphs of the functions f(x) = x^3 + (a + b)^2 + 3x − 4 and g(x) = (x − 3)^2 + 1 touch. Express a in terms of b.



the solution at the back of the book says a =(65 – 8h)/8


How should I be looking at this question??

x^3 + (a+b)x^2 + 9x + 6 = 0 is not true!
You should set x^3 + (a + b)x^2 + 3x − 4 = (x − 3)^2 + 1
That is x^3 + (a+b)x^2 + 3x - 4 = x^2 -6x+9+1
x^3 + (a+b-1)x^2 +9x -14=0

I agree with Dr Peterson's definition of touch.

According to WA (Look here) a+b - 1 =57/8, so a+b = 65/8 or a= 65/8 - b = (65-6b)/8

I graphed both function (using a+b=65/8) and they do not touch but rather they intersect. Check out here

 

[Dr.Peterson]

Entered correctly (without the typo from post #3), they do touch: see here.

What we still lack is a good way to solve the problem based on what you have learned. I'm hoping you will tell us that you know some calculus, but even then I haven't yet seen an easy method. An answer can reasonably be checked without calculus, if necessary, by observing that the difference between the functions touches the x-axis.