Straight line tangents

I've been solving questions about linear tangents to curves and I noticed that when you create a simultaneous equation with a linear tangent you get a repeated root if it is tangent to a quadratic (and this is the only solution) and a repeated root (and one other root) when it is tangent to a cubic. {EDIT 1: Thinking about this a little more, I've realised that you can also have triple repeated roots where the line crosses the curve but has the same gradient as the curve where it crosses.}

I was wondering if the number of repeated roots for a straight-line tangent to a curve is always two, and I thought about quartic curves and I think the roots you can get are:

-2 equal roots that are real (tangent) and two complex roots

-4 real roots: 2 are equal (tangent) and 2 different real rots (where the line crosses the curve)

-2 pairs of equal roots (when the line is tangent to the curve at two points)

Please correct me if I am wrong, but I think that with a straight-line tangent it always just has a double root at the point where it is tangent, and it may or may not have other roots...

But I have also only thought about this using sketches of cubic and quartic curves. I was wondering if there was another way to see why this would be the case?

Also, what would happen if you get a triple root? What does this mean?

EDIT 2: Please scrap all of what I said above! I'm leaving it in just to show my thought process and general question, but after some more investigation with curves, my question has changed a bit... See below!

So I now see that whenever there is a repeated root to an even power, the line is tangent to the curve. When there is a repeated root with an odd power, the line and curve meet and have the same gradient at this point, but they cross. So my questions now are:

  1. What is the significance of having a double root, as opposed to a quadruple root, when the line meets the curve (e.g. a curve defined by a quartic polynomial)? Does it have something to do with the second and third order derivatives too?

  2. What is the significance of complex roots to these equations? Specifically, what would be the difference between a tangent to a quartic where there is a single, real, repeated root to the power of four in the simultaneous equation as opposed to a single real root to the power of two, and a set of complex conjugate roots? My current idea of this is that with a single real root to the power of four, as you diverge away from the tangent point, the distance between the line and the curve keeps on growing, but each single complex conjugate root pair adds another dip of the curve towards line. If that is the case, then what is the significance of the real and imaginary components of the complex pair? I think the real component is the actual x value at which there is the closest PERPENDICULAR distance (I think! Please correct me on this one) between the line and the curve (i.e. the shortest distance following a normal to the tangent, as this is the actual closest distance) as opposed to the shortest vertical distance...) But then I am not sure what the significance of the imaginary part is. What difference does it make if, for instance, the conjugate pair is $-1/+-/2i$ as opposed to $-1/+-/7i$??

Tangents which are curves

This I am not sure about at all. Does it follow the same principles that:

  • Even power repeated root means tangent

  • Odd power repeated root meant same gradient but crossing

  • Extra real roots mean intersection
  • Extra complex conjugate roots means shortening of distance between the two curves?

Thinking about it now, when you solve the simultaneous equations and you set the two functions (i.e. y values) equal to each other, you literally create a distance function and you are trying to find where the distance is zero. Other than the fact that this means solutions to the equation show where the functions meet, whether tangent or intersection, i'm not sure what the significance of this is in my consideration. And it makes me a little more confused as to whether the shortest distance represented by a complex conjugate root pair is the vertical distance or the perpendicular distance. On the one hand, if it were the perpendicular distance, I am essentially imagining tilting the coordinate system so that the line (or lower power curve) is forming the x axis. The problem with it being the perpendicular distance though, is that this shortest distance follows a line which cuts the two curves (or line and curve) at different values of x. Which, then, is the solution that we find when solving the simultaneous equation? The vertical distance would solve this issue of the discrepancy of the x coordinate, and I suppose in that case the 'shortest actual distance' function would be more complicated...

Unless, the 'base' line-i.e. x axis-- was formed by whichever function you subtract. E.g. if you have the curve f(x) and line g(x) and you form the distance function f(x)-g(x)=0, then maybe the distance line must be perpendicuular to the subtracted function, g(x), and not the first function, f(x), and the distance line must only be perpendicular to g(x) and the solution is the x value on f(x) that gives the shortest distance to any point on the line g(x)...

I apologise for the length of this post and hope I have made myself sufficiently clear! Thank you in advance for any help.


First, you've done something great: you've looked at a problem that probably wasn't in your textbook and analyzed it to death and made mistakes and corrected them. That's terrific!

OK, on to answer your questions (and to counter-answer a few).

First, the notion of "tangent" that you're using isn't generally called (by folks doing calculus) "tangent". Your idea of a line $L$ be tangent to a curve $C$ is that $L$ and $C$ share a point $P$ and at the point $P$ they have the same slope and near $P$, the curve $C$ is on one side of $L$. That's almost right...except that the "one side of" clause isn't included in the definition. So we often say that if $f$ is a function whose graph is a curve $C$, and $L$ and $C$ meet at $P$ and have the same slope at $P$, then "$L$ is the tangent to $f$ at $P$", whether $L$ and $C$ cross there or not.

Still, for polynomials it's an interesting and useful concept.

Let me switch to talking about functions.

If $f(x)$ is a polynomial function and $g(x) = mx + b$ is a line (i.e., $m$ is nonzero), and $f(a) = g(a)$ for some $a$, then we say that $f$ and $g$ intersect, or "agree to order zero"; if $f(a) = g(a) = 0$, we say that they have a shared root. If the slopes of $f$ and $g$ are the same, then we say that they "agree to first order." In this case, the equation $f(x) - g(x) = 0$ has (at least) a double-root at $a$, and indeed, for polynomials you could call this a "definition" of the derivative: the linear function $g$ with the property that $f(x) - g(x)$ has a double root at $a$ [or the slope of that line] could be called the derivative of $f$ at $a$.

The root could have higher order; it could be a triple root, or a fourth root, etc. And as you have observed, for odd-count repeated roots (when $f$ is polynomial), this indicates a crossing at $x = a$, while for even-count repeated roots, it indicates that near $x = a$, the graph of $f$ lies on one side of its tangent. By the time you've studied the second derivative and the Mean Value Theorem, you should be able to prove this.

What about those extra roots? Well, if you're willing to consider complex numbers, things actually get easier. Every degree-$n$ complex polynomial has $n$ roots, at least when you count repeated roots the appropriate number of times. So if $f$ is a cubic, when you look at $$ f - g $$ you've still got a cubic. And if they meet and have the same slope at $x = a$, then you get a double-root there...so there has to be a third root somewhere else. There's one exception: look at $f(x) = x^3$ and $g(x) = 0$. These meet at $a = 0$, and $f - g$ has a triple root there. So the "extra" root has gotten lumped in with the double-root at the tangency.

Since every real root must also be a complex root, every real degree $n$ polynomial has at most $n$ roots. So if you have a line tangent to the graph of such a polynomial, there are at most $n-2$ "other meetings" between that tangent line and the graph of the polynomial.

To go in a different direction, not every function is a polynomial. There are functions that touch their tangent lines in very odd ways. Consider the function $$ f(x) = \begin{cases} 0 & x = 0 \\ x^2 \sin \frac{1}{x^2} & x \ne 0 \end{cases} $$ Its tangent at the point $x = 0$ is the $x$-axis (which takes some work to show!), but this tangent intersect the graph of $f$ infinitely often in any neighborhood of $x = 0$. Yikes!

There is also a standard example (which I won't write down, because it probably uses things you haven't yet encountered) of a nonconstant function $f$ with the property that $f(0) = 0$ and $f'(0) = 0$, so its tangent line at $0$ is the $x$-axis. But at $x = 0$, the graph of $f$ meets its tangent "to infinite order," i.e., if you approximate $f$ near zero as well as possible with a degree $n$ polynomial, that polynomial will have $n$ roots all at $0$...and this is true for any $n$. We say that the function $f$ is "extremely flat" at $x = 0$. Such functions are actually important for "constructing partitions of unity" (whatever that might mean!) later on, i.e., they're not just wacko examples, but actually represent important phenomena in mathematics.

My answer's now almost as long as your question, and I hope I've addressed most of what you asked.

My personal favorite book for helping folks understand questions like this is Calculus, by Michael Spivak. It might well be better called something like "An introduction to analysis", but whatever you call it, it's a treasure.


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