Cubic roots
Find the location of the point of inflection of this cubic.
Problem
A certain cubic polynomial $y=f(x)$ cuts the $x$-axis at the three points $x=10, 100$ and $1000$. Is this enough information to determine the location of its point of inflection (note that this is not necessarily a stationary point of inflection)? If so, where is this point; if not, why not?
Construct a cubic polynomial which cuts the $x$-axis at $x=10, 100$ and its point of inflection. How many such polynomials are there?
Did you know ... ?
Polynomials have many fascinating properties. A key result of university mathematics is the Fundamental Theorem of Algebra which states that any polynomial of degree $n$ $p(z)= a_nz^n+a_{n-1}z^{n-1}+\dots+a_0$, with $a_n\neq 0$, has precisely $n$, possibly repeated, complex number solutions.
Student Solutions
A polynomial $f(x)$ has a factor $(x-a)$ if and only if $f(a)=0$.
Thus, a polynomial cutting the $x$-axis at $10, 100, 1000$ has factors $(x-10)(x-100)(x-1000)$. This defines a cubic polynomial up to a multiplicative factor.
Thus
$$f(x) = A(x-10)(x-100)(x-1000) = A\left(x^3 -(10+100+1000)x^2 + .. \right)\,,$$
for some constant $A$.
Now, a point of inflection necessarily has $f''(x) = 0$. Only the $x^3$ and $x^2$ terms of a cubic polynomial contributes to its second derivative, so there is no need to expand the polynomial in full to see that
$$
f''(x) = 6Ax-2220A
$$
This is zero at the single point $x = \frac{2220}{6} = 370$.
Therefore the point of inflection for the cubic is at $x=370$, regardless of the choice of $A$.
For the second part, the polynomial must take the form
$$
f(x) = A(x-10)(x-100)(x-a)\quad \mbox{for a constant } a \mbox{ where} \quad f''(a)=0
$$
So, we need to take the second derivative to work out the constraints on $a$. I will keep the form of the factors and use the chain rule to make life simple, although you could expand the brackets first if you wish
$$
f''(x) = 2A\left((x-10)+(x-100)+(x-a)\right)
$$
So,
$$
f''(a) = 2A(2a-110)=0
$$
Since $A$ cannot be zero for a cubic polynomial, we must have $a=55$.
The polynomial must therefore be of the form
$$
f(x) = A(x-10)(x-55)(x-100)
$$
Alternative, quick, method for second part:
From the first part of the question I noticed a generalisation that the point of inflection of a cubic is found at one third of the sum of the roots $r_1+r_2+r_3$. If one of the roots is the point of inflection then
$$
r_1+a+r_2 = 3a
$$
Thus, the point of inflection which is a root is found at one-half of the sum of the other two roots.
Thus, in our special case,
$$
a = \frac{1}{2}(10+100) = 55, \mbox{ as before}.
$$
Isn't maths great!
Teachers' Resources
Why do this problem?
This is a relatively straightforward problem which will encourage students to think about the relationship between the roots of a cubic and its point of inflection. It also challenges them to discover how much information is required to determine a cubic and its properties.
Possible approach
Have students suggest possible general forms for a cubic function. Does this allow us to determine the equation of a cubic function with roots at 10, 100 and 1000? Is there more than one possible equation? Encourage students to sketch the graphs of their equations. Where would you expect the point of inflection to be? Use the opportunity to clarify what the point of inflection is graphically and how it can be determined analytically.
Once it is clear that students understand what a point of inflection is and how it can be found, they can complete the rest of the problem individually or in pairs and compare their results and approaches with others, ensuring they have reached a conclusion they can agree on and can justify. They should consider whether there are any alternative approaches and if any of these are more efficient, elegant or illuminate something different about the relationship between roots and points of inflection.
Key questions
How can a general cubic equation be expressed? Is this determined uniquely by its roots?
What does the point of inflection of a cubic represent? Can you say where it may be located relative to the roots?
Is your solution unique? How do you know? Are there any other approaches to solving the problem?
Possible support
Students may use Desmos or Geogebra to experiment with different cubic equations with the given roots.