Chapter 3 Applications of Differentiation(1/2)

• Extrema on an interval
• The First Derivative Test
• Concavity and the Second Derivative Test

# Extrema on an interval

# Absolute extrema / Global extrema

Let D be the domain of $f$

1. $f(c)$ is the minimum of $f$ on $D$ when $f(c) \le f(x)$ for all $x \in D$.
2. $f(c)$ is the maximum of $f$ on $D$ when $f(c) \ge f(x)$ for all $x \in D$.
3. The minimum and maximum values of $f$ are called the extreme values of $f$.

# Local extrema / Relative extrema

1. $f$ has a local minimum at $c$ if $f(c) \le f(x)$ when $x$ is near $c$.
2. $f$ has a local maximum at $c$ if $f(c) \ge f(x)$ when $x$ is near $c$.

# Thm

If $f$ is continuous on a closed interval $[ a, b]$, then

1. $f$ is bounded on $[ a, b]$,and
2. $f$ attains its minimum $f(c)$ and its maximum $f(d)$ at some $c$ and $d$ in $[a,b]$

# Define

A critical number of $f$ is a number $c$ in the domain of $f$ such that either $f'(c)=0$ or $f'(c)$ does not exist.

# Fermat's Theorem

If $f$ a local maximum or minimum at $c$ and if $f'(c)$ exists, then $f'(c) = 0$

# The First Derivative Test

# Define

Let $f$ be defined on an interval $I$

1. $f$ is increasing on $I$ if for all $x_1,x_2 \in I$
   $x_1<x_2$ implies that $f(x_1)<f(x_2)$
2. $f$ is decreasing on $I$ if for all $x_1,x_2 \in I$
   $x_1<x_2$ implies that $f(x_1)>f(x_2)$

# Thm

Let $f:[a,b]\to \mathbb{R}$ be continuous and $f:(a,b)\to \mathbb{R}$ be differentiable.

1. If $f'(x)>0$ for all $x\in (a,b)$,then $f$ is increasing on $(a,b)$
2. If $f'(x)<0$ for all $x\in (a,b)$,then $f$ is decreasing on $(a,b)$
3. If $f'(x)=0$ for all $x\in (a,b)$,then $f$ is constant on $(a,b)$

# The First Derivative Test

Let $f:[a,b]\to \mathbb{R}$ be continuous and $f:(a,b)\to \mathbb{R}$ be differentiable.

1. If $f'$ changes from positive to negative at $c$,
   then $f$ has a local maximum at $c$.
2. If $f'$ changes from negative to positive at $c$,
   then $f$ has a local minimum at $c$.
3. If $f'$ does not change sign at $c$,
   then $f$ has no local extrema at $c$.

# Concavity and the Second Derivative Test

# Define

Let $f$ be differentiable on an open interval $I$

1. The graph of $f$ is concave upward (凹向上) on $I$
   if $f'$ is increasing on $I$
2. The graph of $f$ is concave downward (凹向下) on $I$
   if $f'$ is decreasing on $I$

# Concavity Test

Let $f:(a,b) \to \mathbb{R}$ is differentiable.

1. If $f''(x)>0 \forall x \in (a,b)$,then the graph of $f$ concave upward on $(a,b)$
2. If $f''(x)<0 \forall x \in (a,b)$,then the graph of $f$ concave downward on $(a,b)$

# Second Derivative Test

Suppose that $f''$ is continuous near $c$.

1. If $f'(c) = 0$ and $f''(c)>0$,then $f$ has a local minimum at $c$.
2. If $f'(c) = 0$ and $f''(c)<0$,then $f$ has a local maximum at $c$.
3. If $f'(c) = 0$ and $f''(c)=0$,then the test fails.

# Define

The point $(c,f(c))$ is called an inflection point (反曲點) of $f$ if the concavity of $f$ changes from upward to downward or from downward to upward at the point.

# Thm

If $(c,f(c))$ is an inflection point , then $f''(c)=0$ or $f''(c)$ does not exist.

# Reference

• 蘇承芳老師 - 微積分甲（一）109 學年度 - Calculus (I) Academic Year 109