Extremums

Suppose $f:[a,b]\rightarrow \mathbb{R}$, and $F=f([a,b])=\Big\{\,f(x)\;|\;x\in [a,b]\,\Big\}$. Both minimum and maximum values are called the extremums.

Maximum

Maximum of the function $f$ is $f(c)$ where $c\in[a,b]$ iff:

$$\forall x \in [a,b],\; f(c)\ge f(x)$$

aka. Global Maximum. Maximum doesn’t exist always.

Local Maximum

A Local maximum of the function $f$ is $f(c)$ where $c\in[a,b]$ iff:

$$\exists \delta\;\;\forall x\,(0<|x-c|<\delta \implies f(c)\ge f(x))$$

Global maximum is obviously a local maximum.

The above statement can be simplified when $c=a$ or $c=b$.

When $c=a$: $$

$$\exists \delta\;\;\forall x\,(0<x-c<\delta \implies f(c)\ge f(x))$$

When $c=b$: $$

$$\exists \delta\;\;\forall x\,(-\delta<x-c<0 \implies f(c)\ge f(x))$$

Minimum

Minimum of the function $f$ is $f(c)$ where $c\in[a,b]$ iff:

$$\forall x \in [a,b],\; f(c)\le f(x)$$

aka. Global Minimum. Minimum doesn’t exist always.

Local Minimum

$$\exists \delta\;\;\forall x\,(0<|x-c|<\delta \implies f(c)\le f(x))$$

Global minimum is obviously a local maximum.

The above statement can be simplified when $c=a$ or $c=b$.

When $c=a$: $$

$$\exists \delta\;\;\forall x\,(0<x-c<\delta \implies f(c)\le f(x))$$

When $c=b$: $$

$$\exists \delta\;\;\forall x\,(-\delta<x-c<0 \implies f(c)\le f(x))$$

Special cases

f is continuous

Then by Extreme Value Theorem, we know $f$ has a minimum and maximum in $[a,b]$.

f is differentiable

• If $f(a)$ is a local maximum: $f'_{\text{+}}(a)\le 0$
• If $f(b)$ is a local maximum: $f'_{\text{-}}(b)\ge 0$
• $c\in(a,b)$ and If $f(c)$ is a local maximum: $f'(c)= 0$
• If $f(a)$ is a local minimum: $f'_{\text{+}}(a)\ge 0$
• If $f(b)$ is a local minimum: $f'_{\text{-}}(b)\le 0$
• $c\in(a,b)$ and If $f(c)$ is a local minimum: $f'(c)= 0$