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Examples of Concavity of a Curve

Here are some examples of concavity of a curve.

The key to finding the concavity of a curve is to differentiate the equation twice, find the stationary point, point of inflection, and the behaviour of the curve at these points.

 

  1. Consider the curve f(x) = x3 + x2 – 2x – 1f'(x) = 3x2 + 2x – 2

    To find the stationary point, make f'(x) = 0. So

    f'(x) = 3x2 + 2x – 2 = 0

    Applying the quadratic equation x = {-b~pm~sqrt{b^2~-~4ac}}/{2a}

    x = {-2~pm~sqrt{2^2~-~4*3*-2}}/{2*3}

    {-2~pm~sqrt{4~+~24}}/6

    {-2~pm~sqrt{28}}/6

    {-2~pm~2sqrt{7}}/6

    {-1~pm~sqrt{7}}/3

    = 0.55 or -1.22

    Now to find the concavity, we find f”(x) = 6x + 2

    The curve is concave upwards if f”(x) > 0 i.e 6x + 2 > 0

    x > -1/3

    The curve is concave downwards if f”(x) < 0 i.e 6x + 2 < 0

    x < -1/3

    There is a possible point of inflection when f”(x) = 0 i.e 6x + 2 = 0

    x = -1/3

    Check if the concavity changes at the possible point of inflection. We’ll try 2 numbers either side of -1/3.

    f”(-1) = 6(-1) + 2 = -4, and f”(0) – 2

    x -1 -1/3 0
    f”(x) -4 0 2
    Curve + 0

    The concavity changes at x = -1/3, hence (-1/3, 0) is the point of inflection.

    point of inflection for cubic function

  2. Consider the curve f(x) = x6f'(x) = 6x5  and  f”(x) = 30x4

    To find the stationary point, make f'(x) = 0. So

    f'(x) = 6x5 = 0     doubleright  x = 0

    To find the concavity of the curve – when f”(x) > 0 i.e. 30x4 > 0, or x > 0. So the curve is concave upwards

    The curve is concave downwards when f”(x) < 0 i.e. 30x4 < 0, or x < 0.

    For inflection f'(x) = 0      doubleright  x = 0

    When x = 0, y = 06 = 0

    Now, is (0, 0) a point of inflection? We need to check if the concavity changes at this point, by looking at a point on either side, and they are -1 and 1.

    f”(-1) = 30(-1)4  = 30,   and f”(1) = 30(1)4  = 30

    x -1 0 1
    f'(x) 30 > 0 0 30 > 0

    Since f”(x) > 0 on either side of x = 0, there is no change in concavity. Hence (0, 0) is not a point of inflection.

    curve sketching for a function with power of 6

  3. Consider the curve f(x) = x5f'(x) = 5x4  and  f”(x) = 20x3

    The curve will concave upwards when f”(x) > 0 i.e. 20x3 > 0 or x > 0, and the curve will concave downwards when f”(x) < 0 i.e. 20x3 < 0 or x < 0.

    To find the possible point of inflection, f”(x) = 0. So 20x3 = 0    doubleright  x = 0

    Now check the concavity of the curve about x = 0.

    x -1 0 0
    f”(x)

    20(-1)3

    = -20 < 0

    0 20(1)3 = 20 > 0
    Curve f”(x) < 0 0 f”(x) > 0

    When x = 0, y = 0.

    Since concavity changes about x = 0, (0, 0) is a point of inflection.

    Now f'(x) = 0   doubleright x4 = 0 or x = 0 is also a stationary point.

    (0, 0) is a horizontal point of inflection.

    point of inflection for function with power of 5

    Note: When a curve has a point which is both a stationary point and a point of inflection, then the point is a horizontal point of inflection.

  4. This is a real life example. The graph below shows the number of insects in a swamp over several years. Describe how the insect population is changing over the years and explain the rate of change.

    concave down curve

    The curve is concave downwards and is decreasing.

    So f'(x) < 0 and f”(x) < 0

    If N is the number of insects, and t is the time in years,

    dN/dt < 0 and d^2N/dt^2 < 0

    As the curve is decreasing the number of insects in the swamp is also decreasing. And since the curve is concave downwards, the rate of growth of the insects in the swamp is decreasing.