# Finding local extrema of functions: 3 mistakes and 3 tips

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 $$x=0$$.

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.

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 $$\pm x^2$$.

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 $$f(x)$$.

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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