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Questions and Answers
For what values of $a$ is the function $f$ given by $f(x) = x^2 + ax + 1$ increasing on $[1, 2]$?
For what values of $a$ is the function $f$ given by $f(x) = x^2 + ax + 1$ increasing on $[1, 2]$?
- $a > 1$ (correct)
- $a = 0$
- $-1 < a < 1$
- $a < -1$
Prove that the function $f$ given by $f(x) = x^3 - 3x^2 + 3x - 100$ is increasing in $R$.
Prove that the function $f$ given by $f(x) = x^3 - 3x^2 + 3x - 100$ is increasing in $R$.
- The function has a relative maximum at $x = 1$
- The derivative of the function is positive for all $x$ in $R$ (correct)
- The function has no critical points
- The function is always above the x-axis
In which interval is $y = x^2 e^{-x}$ increasing?
In which interval is $y = x^2 e^{-x}$ increasing?
- $(-\infty, \infty)$
- $(-2, 0)$ (correct)
- $(2, \infty)$
- $(0, 2)$
Prove that the function $f$ given by $f(x) = \log(\sin x)$ is increasing on $(0, 2)$ and decreasing on $(\pi, 2\pi)$.
Prove that the function $f$ given by $f(x) = \log(\sin x)$ is increasing on $(0, 2)$ and decreasing on $(\pi, 2\pi)$.
Prove that the function $f$ given by $f(x) = \log|\cos x|$ is decreasing on $(0, 2)$ and increasing on $(3\pi, 2\pi)$.
Prove that the function $f$ given by $f(x) = \log|\cos x|$ is decreasing on $(0, 2)$ and increasing on $(3\pi, 2\pi)$.
Let $I$ be any interval disjoint from $[-1, 1]$. Prove that the function $f$ given by $f(x) = x + 1$ is increasing on $I$.
Let $I$ be any interval disjoint from $[-1, 1]$. Prove that the function $f$ given by $f(x) = x + 1$ is increasing on $I$.
For the function f(x) = x^2, what is the analytical definition of being decreasing on an interval?
For the function f(x) = x^2, what is the analytical definition of being decreasing on an interval?
As per the graphical representation in Fig 6.1, which statement is true about the function f(x) = x^2 to the right of the origin?
As per the graphical representation in Fig 6.1, which statement is true about the function f(x) = x^2 to the right of the origin?
What does Definition 1 state regarding a function being increasing on an interval?
What does Definition 1 state regarding a function being increasing on an interval?
When analyzing the function f(x) = x^2, what happens to the height of the graph as we move from left to right along the graph to the left of the origin?
When analyzing the function f(x) = x^2, what happens to the height of the graph as we move from left to right along the graph to the left of the origin?
What is the marginal revenue when x = 15 for the given function?
What is the marginal revenue when x = 15 for the given function?
In the context of increasing and decreasing functions, if a function is decreasing on an interval I, what relationship must hold true between two points within that interval?
In the context of increasing and decreasing functions, if a function is decreasing on an interval I, what relationship must hold true between two points within that interval?
According to Definition 2, when is a real-valued function f considered increasing at a point x0?
According to Definition 2, when is a real-valued function f considered increasing at a point x0?
For the function f(x) = 7x - 3, why is it considered to be strictly increasing on R?
For the function f(x) = 7x - 3, why is it considered to be strictly increasing on R?
Which theorem, studied in Chapter 5, is required to prove the first derivative test for increasing and decreasing functions?
Which theorem, studied in Chapter 5, is required to prove the first derivative test for increasing and decreasing functions?
If f(x) = c for all x in an interval I, what is the characteristic of the function f according to the text?
If f(x) = c for all x in an interval I, what is the characteristic of the function f according to the text?
In the context of the text, when is a function said to be decreasing on an interval I?
In the context of the text, when is a function said to be decreasing on an interval I?
What must be true for a real-valued function to be considered strictly decreasing on an interval I?
What must be true for a real-valued function to be considered strictly decreasing on an interval I?
