Lagrange's Theorem

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If a function f(x) satisfies all the conditions of the mean value theorem in [0, 2], and f(0) = 0, and |f'(x)| ≤ 1 for all x in [0, 2], then which of the following is true?

f(x) = 2x
f(x) = 3 for at least one x in [0, 2]
f(x) ≤ 2 (correct)
|f(x)| ≤ 1

If f(x) = (x - 3)^2 satisfies the mean value theorem in [3, 4], what is the point on the curve where the tangent is parallel to the chord joining (3, 0) and (4, 1)?

(4, 1)
(1, 4)
(7/2, 1/4)
(7/2, 1/2) (correct)

If f(x) and g(x) are defined and differentiable for x ≥ x0, and f(x0) = g(x0), and f'(x) > g'(x) for x > x0, then which of the following is true?

None of these
f(x) = g(x) for some x > x0
f(x) > g(x) for all x > x0 (correct)
f(x) < g(x) for some x > x0

If f is differentiable for all x, and f(1) = -2, and f'(x) ≥ 2 for all x in [1, 6], then which of the following is true?

f(6) ≥ 8 (D) Signup and view all the answers

If c is the real number of the mean value theorem, then which of the following is true?

c = π/4 (C) Signup and view all the answers

If f satisfies the mean value theorem in [0, 2], and f(0) = 0, and |f'(x)| ≤ 1 for all x in [0, 2], then which of the following is true?

f(x) ≤ 2 (C) Signup and view all the answers

If f(x) = (x - 3)^2 satisfies the mean value theorem in [3, 4], what is the point on the curve where the tangent is parallel to the chord joining (3, 0) and (4, 1)?

(7/2, 1/2) (B) Signup and view all the answers

If f(x) and g(x) are defined and differentiable for x ≥ x0, and f(x0) = g(x0), and f'(x) > g'(x) for x > x0, then which of the following is true?

f(x) > g(x) for all x > x0 (C) Signup and view all the answers

If f is differentiable for all x, and f(1) = -2, and f'(x) ≥ 2 for all x in [1, 6], then which of the following is true?

f(6) ≥ 8 (D) Signup and view all the answers

If c is the real number of the mean value theorem, then which of the following is true?

c = π/4 (B) Signup and view all the answers

If f(x) = x^3, then the value of x in the interval [–2, 2] where the slope of the tangent can be obtained by the mean value theorem is

±4/3 (B) Signup and view all the answers

If f(x) = e^x, then the value of c in the mean value theorem for the interval [0, 1] is

log(e - 1) (A) Signup and view all the answers

If f(x) = x^3 - 6ax^2 + 5x, then the value of a for which the tangent to the curve at x = 7 is parallel to the chord that joins the points of intersection of the curve with the ordinates x = 1 and x = 2 is

35/48 (B) Signup and view all the answers

If f(x) satisfies the mean value theorem in [a, b], then which of the following is true?

a ≤ x1 ≤ b (B) Signup and view all the answers

If f(x) = x^3, then the abscissae of the points of the curve where the slope of the tangent can be obtained by the mean value theorem for the interval [–2, 2] are

±4/3 (A) Signup and view all the answers

What is the condition for a function f(x) to be continuous in the closed interval [a, b]?

f(x) is continuous in the closed interval [a, b] and differentiable in the open interval (a, b) (D) Signup and view all the answers

What is the geometric interpretation of the Mean Value Theorem?

The tangent line is parallel to the chord at some point. (B) Signup and view all the answers

What is the role of Rolle's theorem in the proof of the Mean Value Theorem?

It is used to prove the existence of a point where the derivative is zero. (B) Signup and view all the answers

What is the condition for a function f(x) to satisfy the Mean Value Theorem?

f(x) is continuous in the closed interval [a, b] and differentiable in the open interval (a, b) (D) Signup and view all the answers

What can be concluded about a function f(x) that satisfies the Mean Value Theorem?

The function has a tangent line parallel to the chord at some point. (A) Signup and view all the answers

If f(x) is differentiable in [a, b] and f(a) = f(b), then which of the following is true?

f'(x) = 0 at some point in [a, b] (A) Signup and view all the answers

Let f(x) be a differentiable function in [0, 2] with f(0) = 0 and |f'(x)| ≤ 1 for all x in [0, 2]. What can be said about f(x) in [0, 2]?

|f(x)| ≤ x for all x in [0, 2] (D) Signup and view all the answers

If f(x) and g(x) are defined and differentiable for x ≥ x0, and f(x0) = g(x0), what can be said about f(x) and g(x) for x > x0?

f(x) > g(x) for all x > x0 (A) Signup and view all the answers

What is the geometric interpretation of the mean value theorem?

The tangent to the curve is parallel to the secant line (A) Signup and view all the answers

If f(x) is differentiable in [a, b] and f'(x) ≥ 0 for all x in [a, b], what can be said about f(x) in [a, b]?

f(x) is increasing in [a, b] (B) Signup and view all the answers

If f(x) is a differentiable function in [a, b] and f'(x) ≥ 0 for all x in [a, b], then what can be said about f(x) in [a, b]?

f(x) is non-decreasing in [a, b] (A) Signup and view all the answers

What is the geometric interpretation of the Mean Value Theorem?

The theorem states that there exists a point where the derivative of a function is equal to the slope of the secant line. (B) Signup and view all the answers

If f(x) and g(x) are defined and differentiable for x ≥ x0, and f(x0) = g(x0), and f'(x) > g'(x) for x > x0, then what can be said about f(x) and g(x) for x > x0?

f(x) ≥ g(x) for all x > x0 (A) Signup and view all the answers

What is the role of Rolle's theorem in the proof of the Mean Value Theorem?

Rolle's theorem is used to prove that there exists a point where the derivative of a function is equal to the slope of the secant line. (C) Signup and view all the answers

If f(x) satisfies the Mean Value Theorem in [a, b], and f(a) = f(b), then what can be said about f(x) in [a, b]?

f(x) has a local maximum or minimum in [a, b] (A) Signup and view all the answers

Mean Value Theorem

The Mean Value Theorem states that for a function f(x) continuous on a closed interval [a, b] and differentiable on the open interval (a, b), there exists a point c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

Geometric Interpretation of Mean Value Theorem

The Mean Value Theorem guarantees the existence of a point on the graph of a function where the tangent line is parallel to the secant line connecting the endpoints of the interval.

Condition for Mean Value Theorem

The condition for a function f(x) to satisfy the Mean Value Theorem is that it must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b).

Conclusion of Mean Value Theorem

When a function f(x) satisfies the Mean Value Theorem, we can conclude that there is a point on the graph where the tangent line is parallel to the secant line connecting the endpoints of the interval.