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 has a zero derivative

The value at which the function does not exist

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

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

What is a relative minimum value of a function?

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

The value at which the function has a zero derivative

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

The value at which the function does not exist

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

When the function has a zero derivative at that number

When the function has a non-existent derivative at that number

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

When the function does not exist at that number

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

$0$ Signup and view all the answers

What is a critical value of a function?

A value where the derivative of the function is zero or non-existent Signup and view all the answers

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

$-1$, $3$ Signup and view all the answers

What is the first derivative of the function 𝑓(𝑥) = $3𝑥^2 - 6𝑥 - 9$?

$6𝑥 - 6$ Signup and view all the answers

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

0 Signup and view all the answers

What is the derivative of the absolute value function 𝑔(𝑥) = |𝑥|?

It does not exist Signup and view all the answers

For what value of 𝑐 is the absolute value function 𝑔(𝑥) = |𝑥| not differentiable?

0 Signup and view all the answers

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 Signup and view all the answers

What type of function is 𝑓(𝑥) = $3𝑥^2 - 6𝑥 - 9$ on the interval [a, b] if 𝑓'(𝑥) ≥ 0 for all 𝑥 in (a, b)?

Strictly increasing Signup and view all the answers

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

Decreasing Signup and view all the answers

In Theorem 4.4, if 𝑓′(𝑥) ≥ 0 as 𝑥 approaches 𝑐 from the left and 𝑓′(𝑥) ≤ 0 as 𝑥 approaches 𝑐 from the right, what can be concluded about 𝑓(𝑐)?

It's a relative maximum value Signup and view all the answers

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 Signup and view all the answers

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

When it's either an absolute maximum or an absolute minimum Signup and view all the answers

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

A strictly increasing function Signup and view all the answers

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

A strictly decreasing function Signup and view all the answers

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

${x | x \neq k\pi, k \in \mathbb{Z}}$ Signup and view all the answers

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

$[0, +\infty)$ Signup and view all the answers

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

$f(x) = e^x$ Signup and view all the answers

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

$x \neq 0$ Signup and view all the answers

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

$(-\infty, -1] \cup [1, +\infty)$ Signup and view all the answers

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

$f(x) = e^x$ Signup and view all the answers

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

$[0, +\infty)$ Signup and view all the answers

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

$f(x) = e^x$ Signup and view all the answers

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

$[0, +\infty)$ Signup and view all the answers

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

$f(x) = e^x$ Signup and view all the answers

What is the domain of the function resulting from the addition (𝐹𝐹 + 𝐺𝐺)(𝑥𝑥)?

dom(𝐹𝐹) ∩ dom(𝐺𝐺) Signup and view all the answers

What is the domain of the function resulting from the subtraction (𝐹𝐹– 𝐺𝐺)(𝑥𝑥)?

dom(𝐹𝐹) ∩ dom(𝐺𝐺) Signup and view all the answers

What is the domain of the function resulting from the composition (ℎ𝑥1 ∘ ℎ𝑥2)(𝑥)?

dom(ℎ𝑥2) ∩ {𝑥|ℎ𝑥2(𝑥) ∈ dom(ℎ𝑥1)} Signup and view all the answers

What is the domain of the function resulting from the division (� �)(𝑥)?

dom(� �) = dom(𝑓𝑓) ∩ dom(𝑔𝑔){𝑥|𝑔𝑔(𝑥) = 0} Signup and view all the answers

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

[3, +∞) Signup and view all the answers

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

(−∞, −1] ∪ [2, +∞) Signup and view all the answers

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

[3, +∞) Signup and view all the answers

GE1703 Curve Sketching Minimums and Maximums: Definition 4.1

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Questions and Answers

What is a relative maximum value of a function?

What is a relative minimum value of a function?

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

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

What is a critical value of a function?

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

What is the first derivative of the function 𝑓(𝑥) = $3𝑥^2 - 6𝑥 - 9$?

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

What is the derivative of the absolute value function 𝑔(𝑥) = |𝑥|?

For what value of 𝑐 is the absolute value function 𝑔(𝑥) = |𝑥| not differentiable?

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

What type of function is 𝑓(𝑥) = $3𝑥^2 - 6𝑥 - 9$ on the interval [a, b] if 𝑓'(𝑥) ≥ 0 for all 𝑥 in (a, b)?

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

In Theorem 4.4, if 𝑓′(𝑥) ≥ 0 as 𝑥 approaches 𝑐 from the left and 𝑓′(𝑥) ≤ 0 as 𝑥 approaches 𝑐 from the right, what can be concluded about 𝑓(𝑐)?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What is the domain of the function resulting from the addition (𝐹𝐹 + 𝐺𝐺)(𝑥𝑥)?

What is the domain of the function resulting from the subtraction (𝐹𝐹– 𝐺𝐺)(𝑥𝑥)?

What is the domain of the function resulting from the composition (ℎ𝑥1 ∘ ℎ𝑥2)(𝑥)?

What is the domain of the function resulting from the division (� �)(𝑥)?

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

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

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

What is the domain of the function 𝑄𝑄 resulting from csc�√� �2 − � − 2�?

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

What is the domain of 𝑃� � resulting from √� �2 − � − 2�?

What is the domain of �(�1 � �2 )(�)?

What is the domain of � resulting from csc�√� �2 − � − 2�?

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