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c Determine the point on the parabola f (x) = (x  6)^{2}, at which the tangent is parallel to the chord joining the points (6, 0) & (7, 1). c

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c We verify the "Mean Value theorem" for a function f (x): c

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c Which of the following functions satisfies the mean value theorem? c

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c Determine the point on the parabola f (x) = (x  7)^{2}, at which the tangent is parallel to the chord joining the points (7, 0) & (8, 1). c

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c Find the point on the parabola y = (x + 3)^{2}, at which the tangent is parallel to the chord of the parabola joining the points ( 3, 0) & ( 4, 1). c

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c Find the point on the graph of f (x) = x^{3}, at which the tangent is parallel to the chord joining the points (1, 1) & (3, 27). c

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c Find the point at which the tangent to the curve f(x) = x^{2}  6x + 1 is parallel to the chord joining the points (1,  4) & (3, 8). c

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c Find the point on the curve y = 12(x + 1)(x  2) in the interval [1, 2], at which the tangent is parallel to the x  axis. c

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c State whether the function f (x) = 3x^{2}  2 on [2, 3] satisfies the mean value theorem. c

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c State whether the function f(x) = ln x on [1, 2] satisfies the mean value theorem. c

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c State whether the function f (x) = sin x  sin 2x, x ∈ [0, π] satisfies the mean value theorem. c

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c Find the value of c that satisfies the conclusion of the mean value theorem for f (x) = x^{3}  2x^{2}  x + 3 on [0, 1]. c

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c If f (x) = (x 2) lnx, then there exists at least one value of x in (1, 2) such that x ln x + x = c

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c Find the value of c that satisfies the conclusion of the mean value theorem for the function f (x) = ln x in [1, e]. c

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c Find the value of c that satisfies the conclusion of the mean value theorem for the function f (x) = ex in [0, 1]. c

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c Find the value of c that satisfies the conclusion of the mean value theorem for the function f (x) = sin x in [0, π]. c

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c Find the value of c that satisfies the conclusion of the mean value theorem for the function f (x) = cos x in [ π2 , π2]. c

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c If the value of c that satisfies the conclusion of the mean value theorem for the function f (x) = tan x in [π4, π4], then sec^{2} c = c

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c Select the statement that explains why the mean value theorem does not apply to the function f (x) =  x  in the interval [1, 1]. c

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c Select the statement that explains why the mean value theorem does not apply to the function f (x) = tan x in the interval [0, π]. c

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c Select the statement that explains why the mean value theorem does not apply to the function f (x) = e1x in the interval [1, 1]. c

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c If f (x) = e1x for x ∈ [1, 0) ∪ (0, 1] and = 1 for x = 0 , then c

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c Which statement explains why the mean value theorem does not apply to the function f (x) = xx in the interval [2, 2]? c

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c Every function f (x) that follows the hypotheses of the mean value theorem in [a, b] is c

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c Identify the condition necessary for a function f (x) to satisfy the mean value theorem in the closed interval [a, b]. c

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c If y = f (x) is a continuous curve from the point A (a, f (a)) to the point B (b, f (b)) which has unique tangent at every intermediate point between A and B, then c

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