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Calculus

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If f(x) has zeros at x=a, and x=b, the x coordinate of the turning point between x= a and x=b is 1/2(a+b).
PLease help i don't understand this!

  • Calculus - ,

    is it some times true or always true?? that's the question

  • Calculus - ,

    Turning point is the point at which the slope of the graph changes direction, i.e. from positive to negative or vice versa.

    This happens when dy/dx=0.

    In the case of a quadratic function,
    f(x)=ax²+bx+c,
    dy/dx=2ax+b=0, or this happens when
    x=-b/2a

    It turns ou5 that the real zeroes of the quadratic function are at
    x1,x2=(-b±sqrt(b²-4ac))/2a
    and (x1+x2)/2 = -b/2a.

    So, yes, the "turning point" where the function is a maximum/minimum happens at the average of the two zeroes. However, this is true only for the case of the quadratic equations, and is not generally true for all functions.

  • Calculus - ,

    but what if its not a quadratic function?.. isn't it possible to have a function that has two zeros but is not a quadratic.. for example the first zero is passed by a cubic kind of curve that is connected to a straight line going to the other zero? so would it be quintic, so does the same thing apply?

    And is the question always true, sometimes true, or never true?

  • Calculus - ,

    "is not generally true for all functions. "
    means that it is possible that dy/dx=0 at the average of two roots, but in general it is not true.

    Example when it is true:
    sin(x)=0 at x=0 and x=π.
    dsin(x)/dx=0 at x=π/2

    Example when it is not true:
    y=x(x-1)(x-2)=x³-3x²+2x
    y(0)=y(1)=y(2)=0
    dy/dx=0 at x=1±(√3)/3

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