Posted by AAron on Thursday, March 1, 2012 at 6:58pm.
is it some times true or always true?? that's the question
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.
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?
"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