Rolles Theorem

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What is a necessary condition for Rolle's theorem to be valid?

f(a) ≠ f(b)

f(x) is differentiable in the closed interval [a, b]

f(x) is continuous in the open interval ]a, b[

f(x) is continuous in the closed interval [a, b] and f(x) is differentiable in the open interval ]a, b[ (correct)

What can be said about the function f(x) if it is constant in the interval [a, b]?

The function is increasing at x = a

The function has a minimum value at x = a

f'(x) = 0 for all x ∈ [a, b] (correct)

The function has a maximum value at x = c

What happens to the function f(x) as x takes values slightly greater than a, according to Rolle's theorem?

It always decreases

It remains constant

It always increases

It either increases or decreases (correct)

What is the condition for Rolle's theorem to be applicable?

The function is differentiable in the interval [a, b] Signup and view all the answers

What can be said about the slope of the tangent at x = c, according to Rolle's theorem?

The slope is zero Signup and view all the answers

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 Signup and view all the answers

What is the geometric interpretation of Rolle's theorem?

The tangent is parallel to the x-axis at x = c Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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] Signup and view all the answers

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) Signup and view all the answers

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 Signup and view all the answers

What is the geometric interpretation of Rolle's theorem?

The tangent to the curve at some point is parallel to the x-axis Signup and view all the answers

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] Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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) Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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) Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

Rolles Theorem

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Questions and Answers

What is a necessary condition for Rolle's theorem to be valid?

What can be said about the function f(x) if it is constant in the interval [a, b]?

What happens to the function f(x) as x takes values slightly greater than a, according to Rolle's theorem?

What is the condition for Rolle's theorem to be applicable?

What can be said about the slope of the tangent at x = c, according to Rolle's theorem?

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?

What is the geometric interpretation of Rolle's theorem?

What is the geometric interpretation of the portion AB of the curve y = f(x) in the interval x = a to x = b?

What can be said about the slope of the tangent at x = c if f(x) has a maximum value at x = c?

What is the condition for the minimum value of f(x) to exist 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]?

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?

What is the geometric interpretation of Rolle's theorem?

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?

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?

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)?

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?

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?

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)?

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?

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?

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?

What is the condition for the existence of a maximum value of f(x) in the interval [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)?

What is the relationship between the slope of the tangent at x = c and the derivative of f at x = c?

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[?

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?

What is the geometric interpretation of the portion AB of the curve y = f(x) in the interval x = a to x = b?

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)?

What is the necessary condition for the slope of the tangent to be zero at x = c?

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