Marginal Revenue and Increasing/Decreasing Functions Quiz

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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 $R$.

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?

$(-\infty, \infty)$

$(-2, 0)$ (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 $(\pi, 2\pi)$.

Decreasing on $(0, 2)$ and increasing on $(\pi, 2\pi)$ Signup and view all the answers

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 $(3\pi, 2\pi)$ Signup and view all the answers

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 $x = -1$ Signup and view all the answers

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 Signup and view all the answers

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. Signup and view all the answers

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 Signup and view all the answers

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. Signup and view all the answers

What is the marginal revenue when x = 15 for the given function?

$116 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

Which theorem, studied in Chapter 5, is required to prove the first derivative test for increasing and decreasing functions?

Mean Value Theorem Signup and view all the answers

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 Signup and view all the answers

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 $f(x_1) < f(x_2)$ Signup and view all the answers

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 Signup and view all the answers

Marginal Revenue and Increasing/Decreasing Functions Quiz

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