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)
  • (Some say relative instead of local, and absolut instead of global)
  • A function $g$ said to change its sign at $x_0$ if $\exists \epsilon >0:\forall x_1\in (x_0-\epsilon ,x_0),\forall x_2\in (x_0,x_0+\epsilon ),g(x_1)g(x_2)<0$
    • If $\forall x_1\in (x_0-\epsilon ,x_0),g(x_1)>0$ and $\forall x_2\in (x_0,x_0+\epsilon ),g(x_2)<0$, then $g$ changes sign at $x_0$ from positive to negative (from $+$ to $-$).
    • If $\forall x_1\in (x_0-\epsilon ,x_0),g(x_1)<0$ and $\forall x_2\in (x_0,x_0+\epsilon ),g(x_2)>0$, then $g$ changes sign at $x_0$ from negative to positive (from $-$ to $+$).
  • $x_0$ is called a critical point (חשודה כקיצון) 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 $f^{\prime }(x)=0$
  • $x_0$ is called a turning point of $f$ if $f^{\prime }(x_0)=0$ and $f^{\prime }$ changes its sign at $x_0$.

Theorems

Let $f$ be a function that is continuous at a point $x_0$ and differentiable in a punctured neighborhood $I=N_{\epsilon }(x_0)$ of $x_0$.

• (8.3) A global extremum point on $I$ is either a local extremum point of $f$ or an endpoint of $I$
• (Fermat’s theorem) equivalent versions:
  • (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 $f^{\prime }(x_0)=0$)
• If $x_0$ is a local extremum point of $f$, then $x_0$ is a critical point of $f$.
• (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 $x_0$
  • $f^{\prime }(x_0)=0$ (i.e. $x_0$ is a stationary point of $f$)
• (q8.3) if $f$ is monotonic on $I$, and $x_0$ is global extremum point of $f$, then $x_0$ is an endpoint of $I$
• (8.21) (First Derivative Test for Local Extrema)
  • If $f^{\prime }$ changes its sign at $x_0$ from $-$ to $+$, then $x_0$ is a local minimum point of $f$.
  • If $f^{\prime }$ changes its sign at $x_0$ from $+$ to $-$, then $x_0$ is a local maximum point of $f$.
  • If $f^{\prime }$ does not change its sign at $x_0$, then $x_0$ is not a local extremum point of $f$, and:
    • If $x_0$ is a stationary point of $f$, then $x_0$ is an inflection point of $f$.
• (8.23) (Second Derivative Test for Local Extrema) If $f$ is twice differentiable at $x_0$ and $f^{\prime }(x_0)=0$
  • 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 the test is inconclusive.
• (Second Derivative Test for inflection point)
  • If $f$ is twice differentiable at $x_0$ and $f^{\prime \prime }$ changes its sign at $x_0$, then $x_0$ is an inflection point of $f$.
• (Third Derivative Test) If $f$ is three times differentiable at $x_0$ and $f^{\prime \prime }(x_0)=0$ and $f^{\prime \prime \prime }(x_0)\ne 0$, then $x_0$ is an inflection point of $f$.

$f^{\prime \prime }$	$f^{\prime }$	$f$
positive $+$	increasing $\nearrow$	concave up $\cup$
negative $-$	decreasing $\searrow$	concave down $\cap$
changes sign	extremum point	inflection point