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Identifying Turning Points (Local Extrema) for a Function

Key Questions

  • Answer:

    See below.

    Explanation:

    To find extreme values of a function #f#, set #f'(x)=0# and solve. This gives you the x-coordinates of the extreme values/ local maxs and mins.

    For example. consider #f(x)=x^2-6x+5#. To find the minimum value of #f# (we know it's minimum because the parabola opens upward), we set #f'(x)=2x-6=0# Solving, we get #x=3# is the location of the minimum. To find the y-coordinate, we find #f(3)=-4#. Therefore, the extreme minimum of #f# occurs at the point #(3,-4)#.

  • Any polynomial of degree #n# can have a minimum of zero turning points and a maximum of #n-1#. However, this depends on the kind of turning point.

    Sometimes, "turning point" is defined as "local maximum or minimum only". In this case:

    • Polynomials of odd degree have an even number of turning points, with a minimum of 0 and a maximum of #n-1#.
    • Polynomials of even degree have an odd number of turning points, with a minimum of 1 and a maximum of #n-1#.

    However, sometimes "turning point" can have its definition expanded to include "stationary points of inflexion". For an example of a stationary point of inflexion, look at the graph of #y = x^3# - you'll note that at #x = 0# the graph changes from convex to concave, and the derivative at #x = 0# is also 0.

    If we go by the second definition, we need to change our rules slightly and say that:

    • Polynomials of degree 1 have no turning points.
    • Polynomials of odd degree (except for #n = 1#) have a minimum of 1 turning point and a maximum of #n-1#.
    • Polynomials of even degree have a minimum of 1 turning point and a maximum of #n-1#.

    So, in part, it depends on the definition of "turning point", but in general most people will go by the first definition.

  • For a differentiable function #f(x)#, at its turning points, #f'# becomes zero, and #f'# changes its sign before and after the turning points.


    I hope that this was helpful.

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