18 Questions
For what values of $a$ is the function $f$ given by $f(x) = x^2 + ax + 1$ increasing on $[1, 2]$?
$a > 1$
Prove that the function $f$ given by $f(x) = x^3 - 3x^2 + 3x - 100$ is increasing in $R$.
The derivative of the function is positive for all $x$ in $R$
In which interval is $y = x^2 e^{-x}$ increasing?
$(-2, 0)$
Prove that the function $f$ given by $f(x) = \log(\sin x)$ is increasing on $(0, 2)$ and decreasing on $(\pi, 2\pi)$.
Decreasing on $(0, 2)$ and increasing on $(\pi, 2\pi)$
Prove that the function $f$ given by $f(x) = \log|\cos x|$ is decreasing on $(0, 2)$ and increasing on $(3\pi, 2\pi)$.
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 $I$.
$f(x)$ has a relative minimum at $x = -1$
For the function f(x) = x^2, what is the analytical definition of being decreasing on an interval?
x1 < x2 in I ⇒ f(x1) > f(x2) for all x1, x2 ∈ I
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?
The function is increasing for real numbers x > 0.
What does Definition 1 state regarding a function being increasing on an interval?
x1 < x2 in I ⇒ f(x1) < f(x2) for all x1, x2 ∈ I
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?
The height of the graph continuously decreases.
What is the marginal revenue when x = 15 for the given function?
$116
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?
x1 < x2 in I ⇒ f(x1) < f(x2) for all x1, x2 ∈ I
According to Definition 2, when is a real-valued function f considered increasing at a point x0?
When f is increasing in an open interval containing x0
For the function f(x) = 7x - 3, why is it considered to be strictly increasing on R?
Since f(x1) < f(x2) whenever x1 < x2 in R
Which theorem, studied in Chapter 5, is required to prove the first derivative test for increasing and decreasing functions?
Mean Value Theorem
If f(x) = c for all x in an interval I, what is the characteristic of the function f according to the text?
Constant on I
In the context of the text, when is a function said to be decreasing on an interval I?
$x_1 < x_2$ in I implies $f(x_1) < f(x_2)$
What must be true for a real-valued function to be considered strictly decreasing on an interval I?
$f(x)$ is strictly increasing at every point in I
Test your knowledge on marginal revenue when x = 15 and identifying increasing or decreasing functions using differentiation. The quiz includes questions related to a given function f(x) = x^2 and the corresponding graph.
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