Questions and Answers
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)?
Signup and view all the answers
What is a critical value of a function?
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?
Signup and view all the answers
What is the first derivative of the function 𝑓(𝑥) = $3𝑥^2 - 6𝑥 - 9$?
Signup and view all the answers
Which of the following is a critical value of the function 𝑓(𝑥) = $3𝑥^2 - 6𝑥 - 9$?
Signup and view all the answers
What is the derivative of the absolute value function 𝑔(𝑥) = |𝑥|?
Signup and view all the answers
For what value of 𝑐 is the absolute value function 𝑔(𝑥) = |𝑥| not differentiable?
Signup and view all the answers
According to Theorem 4.2, when does a function have an absolute maximum and an absolute minimum?
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)?
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]?
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)?
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?
Signup and view all the answers
When does a number qualify as an absolute extremum value on an interval?
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?
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?
Signup and view all the answers
What is the domain of the function $f(x) = an(x)$?
Signup and view all the answers
What is the range of the function $f(x) = \sqrt{x}$?
Signup and view all the answers
Which function is defined on the interval $(0, +\infty)$?
Signup and view all the answers
What is the domain of the function $f(x) = \frac{1}{x}$?
Signup and view all the answers
What is the range of the function $f(x) = \ln(x)$?
Signup and view all the answers
Which function has a range of $(-\infty, -1] \cup [1, +\infty)$?
Signup and view all the answers
What is the domain of the function $f(x) = \sqrt{x}$?
Signup and view all the answers
Which function has a range of $[0, +\infty)$?
Signup and view all the answers
What is the domain of the function $f(x) = 2^x$?
Signup and view all the answers
Which function is defined on the interval $(-\infty,-1] \cup [1,+\infty)$?
Signup and view all the answers
What is the domain of the function resulting from the addition (𝐹𝐹 + 𝐺𝐺)(𝑥𝑥)?
Signup and view all the answers
What is the domain of the function resulting from the subtraction (𝐹𝐹– 𝐺𝐺)(𝑥𝑥)?
Signup and view all the answers
What is the domain of the function resulting from the composition (ℎ𝑥1 ∘ ℎ𝑥2)(𝑥)?
Signup and view all the answers
What is the domain of the function resulting from the division (� �)(𝑥)?
Signup and view all the answers
What is the domain of the function 𝑅𝑅 resulting from the square root operation?
Signup and view all the answers
What is the domain of the function 𝑃𝑃 resulting from √� �?
Signup and view all the answers
What is the domain of the function 𝐻𝐻 resulting from (�1 � �2 )(�)?
Signup and view all the answers
What is the domain of the function 𝑄𝑄 resulting from csc�√� �2 − � − 2�?
Signup and view all the answers
What is the domain of 𝐹� � resulting from �1 � �2 ?
Signup and view all the answers
What is the domain of 𝑃� � resulting from √� �2 − � − 2�?
Signup and view all the answers
What is the domain of �(�1 � �2 )(�)?
Signup and view all the answers
What is the domain of � resulting from csc�√� �2 − � − 2�?
Signup and view all the answers