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Examples of Increasing and Decreasing Curves

Let’s now look at a few examples of increasing and decreasing curves

  1. Find the value of x for which the curve y = x3 – 3x2 + 27x – 3 is increasing.
    y = f(x) = x3 – 3x2 + 27x – 3
    y’ = f'(x) = 3x2 – 6x + 27

    = 3(x2 – 2x + 9)

    For the function to be increasing f'(x) > 0

    So x2 – 2x + 9 > 0

    This is a quadratic function, and has a >0

    b2 – 4ac =  4 – 36 = -32 (which is < 0)

    Hence f'(x) i.e. x2 – 2x + 9 does not have any real solutions, and the parabola does not meet x-axis. It is above x-axis and always increasing.
    Hence this is a monotonic increasing curve.

  2. Show that y = 2x3 – 12x2 + 24x – 16 is a monotonic curve and also find its stationary point.
    y = 2x3 – 12x2 + 24x – 16
    y’ = 6x2 – 24x + 24

    = 6(x2 – 4x + 4)

    = 6(x – 2)2

    (x – 2)2 > 0 for all x, so x is always positive

    Therefore y’ > 0 for all x. Hence the curve is monotonic increasing.

    For stationary point –

    f'(x) = 0

    6(x – 2)2  = 0

    (x – 2) = 0

    So x = 2.

    At x = 2, y = 2(2)3 – 12(2)2 + 24×2 – 16 = 0

    So (2, 0) is the stationary point.

    Cubic function with stationary point