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anemone
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The roots of a fourth degree polynomial $g(x)=0$ are in an AP (arithmetic progression). Prove that the roots of $g'(x)=0$ must also form an AP.
MarkFL said:My solution:
Without loss of generality, let us horizontally translate the function $g$ such that it is even in our coordinate system:
\(\displaystyle g(x)=(x+a)(x+3a)(x-a)(x-3a)=\left(x^2-a^2\right)\left(x^2-9a^2\right)\)
Hence:
\(\displaystyle g'(x)=2x\left(x^2-9a^2\right)+2x\left(x^2-a^2\right)=4x(x+\sqrt{5}a)(x-\sqrt{5}a)=0\)
We see that the roots of $g'$ are in fact in an AP.
The theory states that if a function $g(x)$ has a constant first derivative, then the roots of its derivative, $g'(x)$, will be in arithmetic progression (AP).
There are several ways to prove this theory. One method is to use the definition of AP and the fact that the derivative of a constant function is 0. Another approach is to use mathematical induction to show that if the first $n$ roots of $g'(x)$ are in AP, then the $(n+1)$th root will also be in AP.
Yes, this theory has applications in various fields such as physics, engineering, and economics. For example, in physics, this theory can be used to predict the motion of an object with constant acceleration. In economics, it can be applied to analyze the rate of change in a market trend.
Yes, this theory can be extended to higher derivatives. In fact, if a function $g(x)$ has a constant $n$th derivative, then the roots of its $n$th derivative, $g^{(n)}(x)$, will be in AP.
The main limitation of this theory is that it only applies to functions with a constant first derivative. Therefore, it cannot be used to analyze the behavior of more complex functions. Additionally, it may not be applicable to discrete functions or functions with discontinuities.