# Traffic jams and 3 common mistakes on turning points

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?

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 $$f(x)$$.

For 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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