Traffic jams and 3 common mistakes on turning points

car-brake-stop-go This post discusses the commonly confused concept of a turning point.

1st mistake: stationary = turning?

Let \(f(x)\) be the position of a moving car in the \(x\)-axis (on a highway if you wish). Then the derivative \(f'(x)\) is the speed (or the length of the velocity vector) of the car.

The solutions of \(f'(x)=0\) are the points where the speed is 0, so the car is stationary. Hence the solutions of \(f'(x)=0\) are called stationary points of the function \(f(x)\).

If a car comes to a stationary point (a stop), it doesn’t mean that the car will make a U-turn. You have certainly been in a traffic jam, where a car stops for a while and then starts moving again in the same direction. Despite this overwhelming real-life practice, stationary points are often (and wrongly) called turning points even at university.

2nd mistake: stationary points = extrema?

graph-Xcubed Another common mistake is to confuse stationary points with local extrema. We have seen hundreds of scripts claiming something like “\(f'(a)=0\), hence \(x=a\) is a minimum (or a maximum)”. The simple counter-example is \(f(x)=x^3\). Indeed, \(f'(x)=3x^2\), so \(x=0\) is a stationary point. However, \(x=0\) is neither a local minimum nor a local maximum.

A stationary point and a local extremum are different concepts. A stationary point is analytically defined as a solution of \(f'(x)=0\). A local extremum is geometrically defined below.

A proper definition of a local maximum

A point \(x=a\) is called a (strict) local maximum of \(f(x)\) if \(f(x)<f(a)\) for all points \(x\neq a\) sufficiently close to the point \(a\). Notice that the inequality \(f(x)<f(a)\) can’t hold at \(x=a\) and is not required over the whole domain of \(f(x)\).

cosine-meets-horizontal-lineFor example, \(f(x)=\cos x\) has a local maximum at \(x=0\), because \(\cos x<1\) for all \(x\neq 0\) over \(-2\pi<x<2\pi\).

A proper definition of a local minimum

Similarly, a point \(x=a\) is called a (strict) local minimum of \(f(x)\) if \(f(x)>f(a)\) for all points \(x\neq a\) sufficiently close to the point \(a\).

The word strict refers to the strict inequality \(f(x)>f(a)\). Non-strict local extrema allow the condition \(f(x)\geq f(a)\) for all \(x\) sufficiently close to the point \(a\). For example, \(x=0\) can be considered as a non-strict local minimum of \(f(x)=\left\{ \begin{array}{l}
x \mbox{ for } x\geq 0,\\ 0 \mbox{ for } x\leq 0. \end{array}\right.\)

Local minima and maxima can be called local extrema. The word extremum means either a minimum or a maximum. A turning point of a function \(f(x)\) is the same concept as a strict local extremum. Indeed, the graph \(y=f(x)\) “turns” (or makes a U-turn) at any strict local extremum of \(f(x)\). However, experts usually say a local extremum, not a “turning point”.

3rd mistake: extreme point = extreme value?

The final common mistake is to confuse extrema with extreme values by writing, for instance, “\(f(x)=x^2+1\) has the minimum \(f(0)=1\)“. The point \(x=0\) is indeed a local minimum of \(f(x)=x^2+1\). However, the value \(f(0)=1\) is called a local minimum value, not a local minimum point. So a point and the value of a function at this point are very different.

  • Riddle 14: for a function \(f(x)\), should any local extremum be a stationary point?
  • How to submit: to write your full answer, simply submit a comment to this post.
  • Hint: the derivative \(f'(a)\) should be well-defined at any stationary point \(x=a\).
  • Warning: justify that any extremum has \(f'(a)=0\) or give a counter-example.
  • Prize: free 1-year access to one of our interactive web tutorials.
  • Restriction: only the first correct public answer will be rewarded.
  • Update: IS has solved the riddle giving the counter-example f(x)=|x|.

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