30 Questions
What is the condition for a function f(x) to be decreasing?
f'(x) < 0
If f(x) = x^2 - 2x, then f(x) is increasing for which value of x?
x > 1
A function f(x) is said to be monotonic on an interval (a, b) if it is either:
increasing or decreasing on (a, b)
If f'(x) = 2(x - 1), then f(x) is decreasing for which value of x?
x < 1
What is the condition for a function f(x) to be increasing?
f'(x) > 0
Let f(x) = 2x^3 - 5x^2 + 3x + 1. Which of the following intervals is the function strictly increasing?
(1, ∞)
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).
Let f(x) = 3x^4 - 4x^3 + 2x^2 + 1. Which of the following intervals is the function strictly decreasing?
(0, 1)
Consider the function f(x) = e^x(1 - x). Which of the following intervals is the function strictly increasing?
(-∞, 1)
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).
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)
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)
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)
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)
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)
A function f(x) is increasing on an interval (a, b) if its derivative f'(x) is:
positive for all x ∈ (a, b)
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)
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)
Let f(x) = x^3 - 3x^2 + 2x. Find the intervals on which the function is increasing?
x ≥ 1
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)
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
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)
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)]
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]
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)
What is the condition for a function f(x) to be monotonically increasing in a defined interval?
f'(x) > 0
If f(x) = x^2 - 4x + 3, what can be concluded about the function?
It is monotonically increasing in (0, ∞) and decreasing in (-∞, 0)
What is the condition for a function f(x) to be monotonically decreasing in a defined interval?
f'(x) < 0
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
What is the graphical analysis interpretation of f'(x) > 0 for all x ∈ (a, b)?
The tangent line is increasing
Determine where a function is increasing or decreasing using its derivative. Learn about the necessary and sufficient conditions for a function to be monotonic in a given interval.
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