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 - AAron, Thursday, March 1, 2012 at 7:14pm
is it some times true or always true?? that's the question
Calculus - MathMate, Thursday, March 1, 2012 at 7:20pm
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,
dy/dx=2ax+b=0, or this happens when
It turns ou5 that the real zeroes of the quadratic function are at
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 - AAron, Thursday, March 1, 2012 at 8:03pm
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 - MathMate, Thursday, March 1, 2012 at 8:55pm
"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:
dy/dx=0 at x=1±(√3)/3