Explain, please?

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  • 6 months ago

    If a function is always increasing, its slope is positive everywhere.

    Take the derivative of the function to get a function for the slope at every point.

    f'(x) = 3ax² + 2bx + c

    That's a parabola. For it to be positive, it has to be above the x-axis. That means it has to never touch the axis (has no real zeros) and it must be upward facing.

    A quadratic has no zeros if the discriminant is negative.

    d = B² - 4AC < 0

    A = 3a

    B = 2b

    C = c

    d = (2b)² - 4(3a)(c) < 0

    4b² - 12ac < 0

    Divide both sides by 4:

    b² - 3ac < 0

    That satisfies the first part making sure it doesn't cross the x-axis (no real zeros).

    In order to be upward facing, the leading coefficient (3a) must be positive.

    3a > 0

    Divide both sides by 3:

    a > 0

    Answer:

    D. b² - 3ac < 0, a > 0

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