Extrema

Extrema (Maxima & Minima)

• $f$ is defined on an interval $I=\text{dom}(f)$

  • Global Extrema
    • Extremum point: $x_0\in I$ is a maximum point (resp. minimum point) on $I$ of $f$ if $\forall (x\in I),f(x_0)\ge f(x)$ (resp. $f(x_0)\le f(x)$)
      • Extremum value: In this case, $f(x_0)$ is called the maximum (resp. minimum) (value) on $I$, and $f$ is said to have a maximum (resp. minimum) (value) on $I$ at a point $x_0$
  • Local Extrema (מקסימום/מינימום מקומי)
    • Local Extremum point: $x_0\in I$ is a local maximum point (resp. local minimum point) at $I$ if there exists a neighborhood $N_{\epsilon }(x_0)$ such that $\forall x\in N_{\epsilon }(x_0),f(x_0)\ge f(x)$ (resp. $f(x)\ge f(x_0)$) (in both cases נקודת קיצון)
      • In this case, $f(x_0)$ is called a local maximum (value) (resp. local minimum (value)) of $f$ at a point $x_0$. (in both cases $f(x_0)$ is also called local extremum value, ערך קיצון)

• $x_0$ is called a critical point of $f$ if either $f^{\prime }(x_0)=0$ or $f^{\prime }(x_0)$ is undefined

  • $x_0$ is called a stationary point of $f$ if $f^{\prime }(x)=0$

Some say relative instead of local, and absolut instead of global

Theorems

• $f$ is defined on an interval $I$ and $x_0\in I$

  • (8.3) A global extremum point on $I$ is either a local extremum point of $f$ or an endpoint of $I$
  • Fermat’s theorem - equivalence forms:
    • (8.4) Let $x_0$ be a local extremum point of $f$. if $f$ is differentiable at $x_0$, then $f^{\prime }(x_0)=0$
    • (8.19) If $x_0$ is a local extremum point of $f$, then, $f$ is not differentiable at $x_0$, xor, ($f$ is differentiable and $f^{\prime }(x_0)=0$)
  • (p93) If $x_0$ is a global extremum of $f$, then at least one of the following is true:
    • $x_0$ is an endpoint of $I$
    • $f$ is not differentiable at $x_0$
    • $f^{\prime }(x_0)=0$ (stationary point)
  • (q8.3) if $f$ is monotonic on $I$, and $x_0$ is global extremum point of $f$, then $x_0$ is an endpoint of $I$
  • if $x_0$ is a stationary point, then
    • if the derivative $f^{\prime }$ changes its sign as it passes through $x_0$ then $x_0$ is local extremum point
    • if the derivative $f^{\prime }$ does not changes its sign as it passes through $x_0$ then $x_0$ is an inflection point

• (8.21) First Derivative Test for Local Extrema - Let $f$ be a function that is continuous at a point $x_0$ and differentiable in a punctured neighborhood $N_{\epsilon }(x_0)$ of $x_0$.

  • If the derivative $f^{\prime }$ changes its sign as it passes through $x_0$:
    • If the sign change is from $-$ to $+$, then $x_0$ is a local minimum point of $f$.
    • If the sign change is from $+$ to $-$, then $x_0$ is a local maximum point of $f$.
  • If there exists a punctured neighborhood of $x_0$ in which the sign of $f^{\prime }$ is constant, then $x_0$ is not an extremum point of $f$.

• (8.23) Second Derivative Test for Local Extrema - if $f^{\prime }(x_0)=0$ and $f^{\prime \prime }(x_0)$ is defined

  • if$f^{\prime \prime }(x_0)\ne 0$, then $x_0$ is a local extremum point of $f$
    • if $f^{\prime \prime }(x_0)>0$, then $x_0$ is a local minimum point of $f$
    • if $f^{\prime \prime }(x_0)<0$, then $x_0$ is a local maximum point of $f$
  • if$f^{\prime \prime }(x_0)=0$, then $x_0$ can be or not be a local extremum point of $f$