Increasing & Decreasing Function

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What is the condition for a function f(x) to be decreasing?

f'(x) = 0

f'(x) ≠ 0

f'(x) < 0 (correct)

f'(x) > 0

If f(x) = x^2 - 2x, then f(x) is increasing for which value of x?

x > 2

x < 1

x > 1 (correct)

x < 2

A function f(x) is said to be monotonic on an interval (a, b) if it is either:

constant on (a, b)

decreasing on (a, b) and increasing on (b, a)

increasing or decreasing on (a, b) (correct)

increasing on (a, b) and decreasing on (b, a)

If f'(x) = 2(x - 1), then f(x) is decreasing for which value of x?

x < 1 Signup and view all the answers

What is the condition for a function f(x) to be increasing?

f'(x) > 0 Signup and view all the answers

Let f(x) = 2x^3 - 5x^2 + 3x + 1. Which of the following intervals is the function strictly increasing?

(1, ∞) Signup and view all the answers

Consider the function f(x) = x^2 - 4x + 3. Which of the following statements is true about the function?

The function is strictly increasing on (1, 2). Signup and view all the answers

Let f(x) = 3x^4 - 4x^3 + 2x^2 + 1. Which of the following intervals is the function strictly decreasing?

(0, 1) Signup and view all the answers

Consider the function f(x) = e^x(1 - x). Which of the following intervals is the function strictly increasing?

(-∞, 1) Signup and view all the answers

Let f(x) = x^3 - 6x^2 + 9x + 2. Which of the following statements is true about the function?

The function is strictly increasing on (1, 3). Signup and view all the answers

What is the necessary condition for a function f(x) to be increasing on an interval (a, b)?

f'(x) > 0 for all x ∈ (a, b) Signup and view all the answers

If f'(x) < 0 for all x ∈ (a, b), what can be concluded about the function f(x) on the interval (a, b)?

f(x) is decreasing on (a, b) Signup and view all the answers

What is the sufficient condition for a function f(x) to be increasing on an open interval (a, b)?

f'(x) > 0 for all x ∈ (a, b) Signup and view all the answers

Let f(x) be a differentiable function on an open interval (a, b). If f'(x) > 0 for all x ∈ (a, b), what can be concluded about the function f(x)?

f(x) is increasing on (a, b) Signup and view all the answers

What is the graphical analysis interpretation of f'(x) > 0 for all x ∈ (a, b)?

The tangent to the graph of f(x) makes an acute angle with the x-axis on (a, b) Signup and view all the answers

A function f(x) is increasing on an interval (a, b) if its derivative f'(x) is:

positive for all x ∈ (a, b) Signup and view all the answers

What is the condition for a function f(x) to be monotonic on an interval (a, b)?

It is either increasing or decreasing on (a, b) Signup and view all the answers

If a function f(x) is differentiable on an open interval (a, b) and its derivative f'(x) = 0 for all x ∈ (a, b), then what can be concluded about the function f(x)?

It is constant on (a, b) Signup and view all the answers

Let f(x) = x^3 - 3x^2 + 2x. Find the intervals on which the function is increasing?

x ≥ 1 Signup and view all the answers

If a function f(x) is increasing on an interval (a, b) and its derivative f'(x) is continuous on (a, b), then what can be concluded about the function f(x)?

It is strictly increasing on (a, b) Signup and view all the answers

If f'(x) ≥ 0 for all x ∈ (a, b), what can be concluded about the function f(x) on the interval (a, b)?

f(x) is monotonically increasing Signup and view all the answers

Let f(x) be a differentiable function on an open interval (a, b). If f'(c) > 0 for some c ∈ (a, b), what can be concluded about the function f(x)?

f(x) is monotonically increasing on some subinterval of (a, b) Signup and view all the answers

Let f(x) be a strictly increasing function on an interval [a, b]. What can be concluded about the inverse function f^-1?

f^-1 is continuous on the interval [f(a), f(b)] Signup and view all the answers

Consider the function f(x) = x^2 - 4x + 3. What can be concluded about the function f(x) on the interval [0, 2]?

f(x) is strictly increasing on the entire interval [0, 2] Signup and view all the answers

Let f(x) be a continuous function on an interval [a, b]. What is the necessary condition for f(x) to be monotonically increasing on the interval [a, b]?

f'(x) ≥ 0 for all x ∈ (a, b) Signup and view all the answers

What is the condition for a function f(x) to be monotonically increasing in a defined interval?

f'(x) > 0 Signup and view all the answers

If f(x) = x^2 - 4x + 3, what can be concluded about the function?

It is monotonically increasing in (0, ∞) and decreasing in (-∞, 0) Signup and view all the answers

What is the condition for a function f(x) to be monotonically decreasing in a defined interval?

f'(x) < 0 Signup and view all the answers

If f(x) is a differentiable function on an open interval (a, b), what can be concluded about the function if f'(x) > 0 for all x ∈ (a, b)?

The function is monotonically increasing Signup and view all the answers

What is the graphical analysis interpretation of f'(x) > 0 for all x ∈ (a, b)?

The tangent line is increasing Signup and view all the answers

Increasing & Decreasing Function

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Questions and Answers

What is the condition for a function f(x) to be decreasing?

If f(x) = x^2 - 2x, then f(x) is increasing for which value of x?

A function f(x) is said to be monotonic on an interval (a, b) if it is either:

If f'(x) = 2(x - 1), then f(x) is decreasing for which value of x?

What is the condition for a function f(x) to be increasing?

Let f(x) = 2x^3 - 5x^2 + 3x + 1. Which of the following intervals is the function strictly increasing?

Consider the function f(x) = x^2 - 4x + 3. Which of the following statements is true about the function?

Let f(x) = 3x^4 - 4x^3 + 2x^2 + 1. Which of the following intervals is the function strictly decreasing?

Consider the function f(x) = e^x(1 - x). Which of the following intervals is the function strictly increasing?

Let f(x) = x^3 - 6x^2 + 9x + 2. Which of the following statements is true about the function?

What is the necessary condition for a function f(x) to be increasing on an interval (a, b)?

If f'(x) < 0 for all x ∈ (a, b), what can be concluded about the function f(x) on the interval (a, b)?

What is the sufficient condition for a function f(x) to be increasing on an open interval (a, b)?

Let f(x) be a differentiable function on an open interval (a, b). If f'(x) > 0 for all x ∈ (a, b), what can be concluded about the function f(x)?

What is the graphical analysis interpretation of f'(x) > 0 for all x ∈ (a, b)?

A function f(x) is increasing on an interval (a, b) if its derivative f'(x) is:

What is the condition for a function f(x) to be monotonic on an interval (a, b)?

If a function f(x) is differentiable on an open interval (a, b) and its derivative f'(x) = 0 for all x ∈ (a, b), then what can be concluded about the function f(x)?

Let f(x) = x^3 - 3x^2 + 2x. Find the intervals on which the function is increasing?

If a function f(x) is increasing on an interval (a, b) and its derivative f'(x) is continuous on (a, b), then what can be concluded about the function f(x)?

If f'(x) ≥ 0 for all x ∈ (a, b), what can be concluded about the function f(x) on the interval (a, b)?

Let f(x) be a differentiable function on an open interval (a, b). If f'(c) > 0 for some c ∈ (a, b), what can be concluded about the function f(x)?

Let f(x) be a strictly increasing function on an interval [a, b]. What can be concluded about the inverse function f^-1?

Consider the function f(x) = x^2 - 4x + 3. What can be concluded about the function f(x) on the interval [0, 2]?

Let f(x) be a continuous function on an interval [a, b]. What is the necessary condition for f(x) to be monotonically increasing on the interval [a, b]?

What is the condition for a function f(x) to be monotonically increasing in a defined interval?

If f(x) = x^2 - 4x + 3, what can be concluded about the function?

What is the condition for a function f(x) to be monotonically decreasing in a defined interval?

If f(x) is a differentiable function on an open interval (a, b), what can be concluded about the function if f'(x) > 0 for all x ∈ (a, b)?

What is the graphical analysis interpretation of f'(x) > 0 for all x ∈ (a, b)?

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