30 Questions
What is a necessary condition for Rolle's theorem to be valid?
f(x) is continuous in the closed interval [a, b] and f(x) is differentiable in the open interval ]a, b[
What can be said about the function f(x) if it is constant in the interval [a, b]?
f'(x) = 0 for all x ∈ [a, b]
What happens to the function f(x) as x takes values slightly greater than a, according to Rolle's theorem?
It either increases or decreases
What is the condition for Rolle's theorem to be applicable?
The function is differentiable in the interval [a, b]
What can be said about the slope of the tangent at x = c, according to Rolle's theorem?
The slope is zero
What can be concluded about the function f(x) at x = c if f(x) increases in the interval a < x < c and then decreases in the interval c < x < b?
The function has a maximum value at x = c
What is the geometric interpretation of Rolle's theorem?
The tangent is parallel to the x-axis at x = c
What is the geometric interpretation of the portion AB of the curve y = f(x) in the interval x = a to x = b?
The curve is continuous and goes from A to B
What can be said about the slope of the tangent at x = c if f(x) has a maximum value at x = c?
The slope is zero
What is the condition for the minimum value of f(x) to exist in the interval [a, b]?
The function is continuous and differentiable in the interval [a, b]
What is the condition for Rolle's theorem to be applicable to a function f(x) defined on [a, b]?
f(x) is continuous on [a, b] and f(a) = f(b)
If a function f(x) satisfies the conditions of Rolle's theorem on [1, 3], then what can be said about the behavior of the function at x = 2?
The function has a critical point at x = 2
What is the geometric interpretation of Rolle's theorem?
The tangent to the curve at some point is parallel to the x-axis
If a function f(x) has two distinct roots in [0, 1], then what can be said about the behavior of the function in this interval?
The function has a maximum or minimum value in [0, 1]
If a function f(x) is defined as f(x) = ax^3 + bx^2 + 11x - 6, then what can be said about the behavior of the function at x = 2?
The function has a critical point at x = 2
If f(x) = ax^3 + bx^2 + 11x - 6 satisfies the conditions of Rolle's theorem on [1, 3], then what can be said about the function f'(2)?
f'(2) = 0
If f(x) = x^3 - 3x + a has two distinct roots in [0, 1], then what can be said about the function f(x) in this interval?
f(x) has a maximum value
If a function f(x) is defined on [a, b] and satisfies the conditions of Rolle's theorem, then what can be said about the tangent to the curve at x = c?
The tangent is parallel to the x-axis
If f(x) is a polynomial of degree n, then what can be said about the number of zeros of f(x) in the interval (0, 1)?
At least one zero
If a function f(x) is defined on [a, b] and has a maximum value at x = c, then what can be said about the slope of the tangent at x = c?
The slope is zero
What can be concluded about the derivative of f at x = c if f is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b) = 0?
f'(c) = 0
What is the geometric interpretation of the portion AB of the curve y = f(x) in the interval x = a to x = b if f increases in the interval a < x < c and then decreases in the interval c < x < b?
The curve has a horizontal tangent at x = c
What is the condition for the existence of a maximum value of f(x) in the interval [a, b]?
f'(c) = 0 for some c in (a, b)
What can be said about the function f(x) if it is differentiable on (a, b) and f'(x) = 0 for some x = c in (a, b)?
f(x) has a maximum or minimum value at x = c
What is the relationship between the slope of the tangent at x = c and the derivative of f at x = c?
The slope of the tangent is equal to the derivative
What is the condition for Rolle's theorem to be applicable to a function f(x) defined on [a, b], given that f(x) is continuous in the closed interval [a, b] and differentiable in the open interval ]a, b[?
f(a) = f(b)
What can be said about the function f(x) at x = c, if f(x) increases in the interval a < x < c and then decreases in the interval c < x < b?
The function has a maximum value at x = c.
What is the geometric interpretation of the portion AB of the curve y = f(x) in the interval x = a to x = b?
The curve has a horizontal tangent at x = c.
What can be said about the function f(x) if it is not constant in the interval [a, b] and since f(a) = f(b)?
The function must either increase or decrease in the interval a < x < b.
What is the necessary condition for the slope of the tangent to be zero at x = c?
The function must have a maximum value at x = c.
This quiz covers the application of Rolle's theorem to determine differentiability of a function at a point. It involves analyzing the limits of the function from both sides to check for differentiability.
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