GE1703 Curve Sketching Minimums and Maximums: Definition 4.1

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What is a relative maximum value of a function?

The value at which the function reaches its highest point in the given interval

What is a relative minimum value of a function?

The value at which the function reaches its lowest point in the given interval

When is a number considered to be a relative extremum value of a function?

When the function reaches its maximum or minimum value at that number

According to Theorem 4.1, if 𝑓𝑓(𝑐𝑐) is a relative extremum value, what is the value of 𝑓𝑓’(𝑐𝑐)?

$0$

What is a critical value of a function?

A value where the derivative of the function is zero or non-existent

In Example: Find the critical values of 𝑓𝑓(π‘₯π‘₯) = $π‘₯^3 - 3π‘₯^2 - 9π‘₯ + 1$, what are the critical values?

$-1$, $3$

What is the first derivative of the function 𝑓(π‘₯) = $3π‘₯^2 - 6π‘₯ - 9$?

$6π‘₯ - 6$

Which of the following is a critical value of the function 𝑓(π‘₯) = $3π‘₯^2 - 6π‘₯ - 9$?

0

What is the derivative of the absolute value function 𝑔(π‘₯) = |π‘₯|?

It does not exist

For what value of 𝑐 is the absolute value function 𝑔(π‘₯) = |π‘₯| not differentiable?

0

According to Theorem 4.2, when does a function have an absolute maximum and an absolute minimum?

When it is continuous on a closed interval

What type of function is 𝑓(π‘₯) = $3π‘₯^2 - 6π‘₯ - 9$ on the interval [a, b] if 𝑓'(π‘₯) β‰₯ 0 for all π‘₯ in (a, b)?

Strictly increasing

If 𝑓′(π‘₯) ≀ 0 for all π‘₯ in (a, b), what type of function is 𝑓(π‘₯) = $3π‘₯^2 - 6π‘₯ - 9$ on the interval [a, b]?

Decreasing

In Theorem 4.4, if 𝑓′(π‘₯) β‰₯ 0 as π‘₯ approaches 𝑐 from the left and 𝑓′(π‘₯) ≀ 0 as π‘₯ approaches 𝑐 from the right, what can be concluded about 𝑓(𝑐)?

It's a relative maximum value

What does Theorem 4.3 state about the behavior of a function when its first derivative is always non-negative on an interval?

It's increasing on that interval

When does a number qualify as an absolute extremum value on an interval?

When it's either an absolute maximum or an absolute minimum

What type of function is defined as strictly increasing on an interval if its values consistently increase and never decrease?

A strictly increasing function

If a function’s values consistently decrease as the input values increase, what type of function is it described as?

A strictly decreasing function

What is the domain of the function $f(x) = an(x)$?

${x | x \neq k\pi, k \in \mathbb{Z}}$

What is the range of the function $f(x) = \sqrt{x}$?

$[0, +\infty)$

Which function is defined on the interval $(0, +\infty)$?

$f(x) = e^x$

What is the domain of the function $f(x) = \frac{1}{x}$?

$x \neq 0$

What is the range of the function $f(x) = \ln(x)$?

$(-\infty, -1] \cup [1, +\infty)$

Which function has a range of $(-\infty, -1] \cup [1, +\infty)$?

$f(x) = e^x$

What is the domain of the function $f(x) = \sqrt{x}$?

$[0, +\infty)$

Which function has a range of $[0, +\infty)$?

$f(x) = e^x$

What is the domain of the function $f(x) = 2^x$?

$[0, +\infty)$

Which function is defined on the interval $(-\infty,-1] \cup [1,+\infty)$?

$f(x) = e^x$

What is the domain of the function resulting from the addition (𝐹𝐹 + 𝐺𝐺)(π‘₯π‘₯)?

dom(𝐹𝐹) ∩ dom(𝐺𝐺)

What is the domain of the function resulting from the subtraction (𝐹𝐹– 𝐺𝐺)(π‘₯π‘₯)?

dom(𝐹𝐹) ∩ dom(𝐺𝐺)

What is the domain of the function resulting from the composition (β„Žπ‘₯1 ∘ β„Žπ‘₯2)(π‘₯)?

dom(β„Žπ‘₯2) ∩ {π‘₯|β„Žπ‘₯2(π‘₯) ∈ dom(β„Žπ‘₯1)}

What is the domain of the function resulting from the division (οΏ½ οΏ½)(π‘₯)?

dom(οΏ½ οΏ½) = dom(𝑓𝑓) ∩ dom(𝑔𝑔){π‘₯|𝑔𝑔(π‘₯) = 0}

What is the domain of the function 𝑅𝑅 resulting from the square root operation?

[3, +∞)

What is the domain of the function 𝑃𝑃 resulting from √� οΏ½?

(βˆ’βˆž, βˆ’1] βˆͺ [2, +∞)

What is the domain of the function 𝐻𝐻 resulting from (�1 � �2 )(�)?

[3, +∞)

What is the domain of the function 𝑄𝑄 resulting from csc�√� οΏ½2 βˆ’ οΏ½ βˆ’ 2οΏ½?

(βˆ’βˆž, βˆ’1] βˆͺ [2, +∞)

What is the domain of 𝐹� � resulting from �1 � �2 ?

[3, +∞)

What is the domain of 𝑃� οΏ½ resulting from √� οΏ½2 βˆ’ οΏ½ βˆ’ 2οΏ½?

(βˆ’βˆž, βˆ’1] βˆͺ [2, +∞)

What is the domain of οΏ½(οΏ½1 οΏ½ οΏ½2 )(οΏ½)?

[3, +∞)

What is the domain of οΏ½ resulting from csc�√� οΏ½2 βˆ’ οΏ½ βˆ’ 2οΏ½?

(βˆ’βˆž, βˆ’1] βˆͺ [2, +∞)

Understand the definition of relative maximum and relative minimum values of a real-valued function 𝑓𝑓 with respect to a point 𝑐𝑐 ∈ ℝ, as per Definition 4.1. Learn when a number is considered a relative maximum or relative minimum value of the function 𝑓𝑓.

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