Critical Points (College Board AP® Calculus AB): Revision Note

Local extrema versus global extrema

What is the difference between local and global extrema?

• The term extremum (plural extrema) refers to maximum and minimum points on the graph of a function

• An extremum can be either local (relative) or global

  • A global extremum is the maximum or minimum for the whole of a function's domain

  • A local or relative extremum is the maximum or minimum within a specific part of a function's domain

• We can say that $f\left(x\right)$ has a

  • global maximum at $x=c$ if

    • $f\left(x\right)\le f\left(c\right)$ for every $x$ in the domain

  • local maximum at $x=c$ if

    • $f\left(x\right)\le f\left(c\right)$ for every $x$ in some open interval around $x=c$

  • global minimum at $x=c$ if

    • $f\left(x\right)\ge f\left(c\right)$ for every $x$ in the domain

  • local minimum at $x=c$ if

    • $f\left(x\right)\ge f\left(c\right)$ for every $x$ in some open interval around $x=c$

• Every global extremum will also be a local extremum

  • However, not all local extrema are global extrema

Critical points

What is a critical point?

• A critical point is a point where the first derivative of a function is

  • equal to zero,

  • or does not exist

    • In the case where $f^{\text{'}}\left(a\right)$ does not exist, the function itself must still be defined at $x=a$ (i.e. $f\left(a\right)$ must exist) in order for $x=a$ to be a critical point

    • E.g. $y=\frac{1}{x}$ does not have a critical point at $x=0$, but $y=x^{\frac{2}{3}}$ does

• All local extrema occur at critical points

  • However, not all critical points are local extrema

What different types of critical points are there?

Local minimums and maximums

• At local minimums and maximums, the derivative is most often equal to zero

  • I.e. for a function $f$, $f^{\text{'}}\left(x\right)=0$ at a local minimum or maximum

  • The tangent at these points is horizontal

• Depending on the function, these can also be global extrema

  • E.g. The graph of $y={\left(x-2\right)}^2$ has a local minimum at (2,0), which is also a global minimum

Points of inflection

• A point of inflection is a point where a graph changes concavity

  • You can read more about concavity in the 'Concavity of Functions' study guide

• Points of inflection are not local extrema

• Only points of inflection where the first derivative is zero are critical points

  • It is possible for points of inflection to exist where the first derivative is not zero

• A point of inflection with the first derivative zero is sometimes referred to as a saddle point

  • E.g. $y=x^3$ has a saddle point at $x=0$

Points where the derivative does not exist

• As well as where the derivative is zero, a critical point can occur at points where the derivative does not exist

  • If the derivative does not exist at a point, the function itself must still be defined at that point to be a critical point

• Consider the function $f$ defined by $f\left(x\right)=\frac{1}{x}$,

  • The function is undefined at $x=0$, so will not have a critical point at $x=0$

• Consider the function $g$ defined by $g\left(x\right)=\sqrt[3]{x}=x^{\frac{1}{3}}$

  • The function is defined at $x=0$ (it has a value of $g\left(0\right)=0$)

  • It has a first derivative of $g^{\text{'}}\left(x\right)=\frac{1}{3}{x}^{-\frac{2}{3}}=\frac{1}{3\sqrt[3]{x^2}}$

  • The first derivative does not exist at $x=0$

  • So $g$ has a critical point at $x=0$

• The image below shows a similar example

• It is also possible for a critical point to exist at a point that is defined but not continuous

  • This means the derivative does not exist at that point

  • The function jumps to a higher or lower value