How do we know the degree of a root which is also a point of inflexion?

[karan000]

Hey guys, I was looking at an exam I did last year and tried to solve a question, which at the time I couldn't do.

Unfortunately I'm running into the same problem I had during the exam, so hear me out on this one

Question:
The graph below has equation y =ax(x-b)(x+c)^d. Write down the values for a, b, c and d.

Okay, so there's an intercept at x=-1, so b=-1. There's another intercept at x=3, so c = -3

So, y = ax(x+1)(x-3)^d

Now here's the problem, the answers simply say "d = 3", and then sub in the point (1,-16) and solve for a,

But at x=3 (where it's a point on inflexion), can't the degree (d) be ANY odd number? ie. 3,5,7,9,11,...,999999?

Because that's how I learned polynomials, a point on inflexion at a root means the degree of the root is odd.

And so that's where I messed up, becausing I couldn't figure what value of 'd' I should use.

I must obviously be missing some sort of concept, so can someone please help me out?

 

[epenguin]

But at x=3 (where it's a point on inflexion), can't the degree (d) be ANY odd number? ie. 3,5,7,9,11,...,999999?

Because that's how I learned polynomials, a point on inflexion at a root means the degree of the root is odd.

And so that's where I messed up, becausing I couldn't figure what value of 'd' I should use.

I must obviously be missing some sort of concept, so can someone please help me out?

You idea is right - it could be any odd number - so far as you have got.

You now have to ask yourself the question, one of the five or so in Polya's 'How to solve it' book

-

"Am I using all the information I'm given?"

 

[karan000]

You idea is right - it could be any odd number - so far as you have got.

You now have to ask yourself the question, one of the five or so in Polya's 'How to solve it' book

-

"Am I using all the information I'm given?"

I'm aware of the the point (1,-16),

So as you already know that b=-1 and c=-3, sub in the point to get:

-16 = a(1)(1+1)(1-3)^d
= 2a(-2)^d

Two varibles makes this unsolvable but using guess and check you'll find 3 is the only correct solution (which just makes the whole question practically ****** and pointless).

I was expecting there to be a way to solve the question without guess and check but I guess there isn't any other way...

 

[epenguin]

OK, you have used the fact that (1, -16) is a point on the function.

But I think you are supposed to use also the fact that it is a special kind of point, a stationary point.

 

[CellCoree]

I would approach this question by first clarifying the definition of a point of inflection and its relationship to the degree of a root. A point of inflection on a graph is a point where the curvature changes from convex to concave or vice versa. This means that the graph changes from curving upwards to curving downwards or vice versa at that point.

Now, let's consider the roots of a polynomial function. The degree of a root is the number of times the root appears as a solution to the polynomial equation. In other words, it is the number of times the graph crosses the x-axis at that root.

So, how do we determine the degree of a root that is also a point of inflection? We need to consider the behavior of the graph near that point. If the graph changes from curving upwards to curving downwards at that point, then the degree of the root must be odd. This is because an odd degree polynomial will have a graph that changes direction at its roots. On the other hand, if the graph changes from curving downwards to curving upwards at that point, then the degree of the root must be even. This is because an even degree polynomial will have a graph that does not change direction at its roots.

In this case, we can see that the graph changes from curving upwards to curving downwards at x=3. Therefore, the degree of the root at x=3 must be odd. This means that d=3 is the correct value to use in the equation. If d were any other odd number, the graph would not have a point of inflection at x=3.

I hope this explanation helps you understand the concept better. As a scientist, it is important to always clarify definitions and consider the behavior of the system in order to arrive at a correct conclusion.