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If f(x) = ax2 + bx + c,then the value of G in the first mean value theorem f(x + h) - f(x) = h f (x + θh) i s _________ .
    Correct answer is '0.5'. Can you explain this answer?
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    Substituting all these values in the first mean value theorem
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    The first mean value theorem states that for a function f(x) that is continuous on the interval [a, b] and differentiable on the interval (a, b), there exists at least one value c in the interval (a, b) such that:

    f'(c) = (f(b) - f(a))/(b - a)

    In this case, we are given that f(x) = ax^2 + bx + c.

    To find the value of G in the first mean value theorem, we need to compute f(x + h) - f(x) and h*f'(x + G*h), and equate them.

    First, let's compute f(x + h) - f(x):

    f(x + h) - f(x) = (a(x + h)^2 + b(x + h) + c) - (ax^2 + bx + c)
    = (ax^2 + 2ahx + ah^2 + bx + bh + c) - (ax^2 + bx + c)
    = 2ahx + ah^2 + bh

    Now, let's compute h*f'(x + G*h):

    f'(x + G*h) = 2a(x + G*h) + b

    h*f'(x + G*h) = h*(2a(x + G*h) + b)
    = 2ahx + 2aGh^2 + bh

    Equating the two expressions, we have:

    2ahx + ah^2 + bh = 2ahx + 2aGh^2 + bh

    Canceling out the common terms, we are left with:

    ah^2 = 2aGh^2

    Dividing both sides by ah^2, we get:

    1 = 2G

    Therefore, G = 1/2.
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    If f(x) = ax2 + bx + c,then the value of G in the first mean value theorem f(x + h) - f(x) = h f (x + θh) i s _________ .Correct answer is '0.5'. Can you explain this answer?
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