Monthly Archives: May 2014

Finding local extrema of functions: 3 mistakes and 3 tips

local-extrema-types This post discusses common mistakes and gives tips how to easily find local extrema of real-valued 1-variable functions.

1st mistake: any local extremum = a stationary point?

This myth has been dethroned in riddle 14. Actually, the function \(f(x)=|x|\) has a local (even global) minimum at \(x=0\), because \(|x|\geq 0\) for all \(x\). However, the derivative \(f'(x)=\left\{ \begin{array}{l} 1 \mbox{ for } x>0,\\ -1 \mbox{ for } x<0 \end{array} \right. \) is undefined at \(x=0\).

graph-Xcubed On the other hand, the function \(f(x)=x^3\) from the previous post has a stationary point at \(x=0\), which is not a local extremum. So the concepts of a stationary point and a local extremum are independent in the sense that one doesn’t follow from another.

1st tip: find all points where the derivative is undefined

The examples above imply that we should study all point where \(f'(x)\) is undefined to find local extrema of \(f(x)\). If the derivative \(f'(x)\) is a fraction, we may start from
finding all points where the denominator of \(f'(x)\) vanishes.

graph-square-root For example, the function \(f(x)=\sqrt{x}\) is well-defined for all \(x\geq 0\), but the derivative \(f'(x)=\frac{1}{2\sqrt{x}}\) is not defined at \(x=0\), where the tangent line to \(y=\sqrt{x}\) is vertical.

2nd mistake: sufficient or necessary conditions of extrema

Recall that a stationary point of a function \(f(x)\) is by definition a solution of the equation \(f'(x)=0\). Here is a theorem stating when a stationary point \(x=a\) is a local extremum.

Sufficient conditions of a local extremum. For a stationary point \(x=a\) of a function \(f(x)\) with a well-defined derivative \(f'(x)\) for all \(x\), if \(f”(a)<0\) then \(x=a\) is a local maximum, if \(f”(a)>0\) then \(x=a\) is a local minimum.

The conditions above are sufficient, but are not necessary as the example of \(f(x)=|x|\) at \(x=0\) shows. If \(f”(a)=0\), then more analysis is needed. This is a typical question of singularity theory, which is richer for more than 1 variable.

2nd tip: check your rule for basic shapes of \(\pm x^2\).

local-extrema-types Many students often forget which inequality \(f”(a)<0\) or \(f”(a)>0\) corresponds to a local maximum or a local minimum. The simple trick is to remember the basic shapes of \(x^2\) and \(-x^2\). Namely, the positive parabola \(f(x)=x^2\) has \(f”(0)=2>0\) and a local minimum at \(x=0\). Similarly, the negative parabola \(f(x)=-x^2\) has \(f”(0)=-2<0\) and a local maximum at \(x=0\).

3rd mistake: does \(f”(a)=0\) mean that \(x=a\) is a point of inflection?

We have seen dozens of student scripts wrongly claiming that “\(f”(0)=0 \Rightarrow x=0\) is a point of inflection”. Here is a proper geometric definition of a point of inflection of \(f(x)\): if the graph \(y=f(x)\) goes from one side of its the tangent line at \(x=a\) to another side in a small neighbourhood of \(x=a\), then \(x=a\) is a point of inflection.

Analytically, if \(L(x)=f(a)+f'(a)(x-a)\) is the tangent line to \(y=f(x)\), then the difference \(f(x)-L(x)\) changes its sign at a point of inflection \(x=a\). For instance, at a stationary point \(x=a\) the tangent line \(L(x)=f(a)\) is horizontal and the same difference \(f(x)-f(a)\) keeps its sign around \(x=a\). Hence a point of inflection can not be a local extremum of \(f(x)\).

graph-Xcubed For example, \(x=0\) is a point of inflection of \(f(x)=x^3\), because \(y=x^3\) intersects the tangent line \(y=0\) and goes from the lower half-plane to the upper half-plane.

3rd tip: use well-known inequalities for justifying extrema

The function \(f(x)=x^4\) has a local minimum, but not a point of inflection at \(x=0\). The sufficient conditions from the theorem above are not satisfied as \(f'(0)=0=f”(0)\). However, the justification is even easier: \(x=0\) is a global minimum of \(f(x)=x^4\) over all real \(x\) as \(x^4\geq 0\).

  • Riddle 15: state conditions when \(x=a\) is a point of inflection of \(f(x)\).
  • How to submit: to write your full answer, simply submit a comment.
  • Hint: give sufficient conditions in derivatives of \(f(x)\) without proof.
  • Warning: not all conditions are equalities, \(f'(a)=0\) isn’t enough.
  • Prize: free 1-year access to one of our interactive web tutorials.
  • Restriction: only the first correct public answer will be rewarded.

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