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).

• 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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