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A polynomial function is an expression consisting of variables and coefficients structured as a combination of terms. These terms involve products of variables raised to power (exponents) and their related coefficients. A general polynomial function looks something like:

• Constant term: A number by itself, such as 5.
• Linear term: A term like 2x, where the variable is raised to the first power.
• Quadratic term: A term like 3x², where the variable is squared.
• Cubic term: A term like x³, where the variable is cubed, and so on.

Understanding polynomial functions involves recognizing these components and patterns. In this exercise, the function given is a cubic polynomial: \[ f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \] Here, the variable is \(x\), and the polynomial combines linear, quadratic, and cubic terms with varying coefficients. Each term contributes to the overall shape and behavior of the polynomial on a graph.

The derivative of a function is a critical concept in calculus and mathematical analysis. It represents the rate at which a function is changing at any given point. Differentiating a function helps to locate where the function increases, decreases, achieves a peak, or hits a valley. This is exactly what you need when you want to find critical points.
When finding the derivative of a polynomial function, you apply the power rule which states that you bring down the exponent as a multiplier and reduce the exponent by one:

• The derivative of \(x^n\) is \(nx^{n-1}\).
• For example, the derivative of \(2x^3\) is \(6x^2\).

In the problem at hand, differentiating the function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \) gives:\[ f'(x) = 6x^2 - 18ax + 12a^2 \] This new equation, \(f'(x)\), is crucial for determining where the function rises or falls, by solving \(f'(x) = 0\) to find the critical points.

A quadratic equation is a polynomial equation of the second degree, typically presented in the form \( ax^2 + bx + c = 0 \). Solving these equations involves finding the values of \(x\) where this equality holds true. This may result in two, one, or no real solutions, depending on the discriminant, \(b^2 - 4ac\):

• If positive, two distinct solutions exist.
• If zero, one real solution exists.
• If negative, no real solutions exist.

In the derivative \( f'(x) = 6x^2 - 18ax + 12a^2 \), after setting it to zero, simplifying gives:\[ x^2 - 3ax + 2a^2 = 0 \] This is recognized as a quadratic equation. To find the roots or solutions, one uses the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Plugging in the values \(b = -3a\), \(a = 1\), and \(c = 2a^2\), simplifies to find solutions for \(x = 2a\) or \(x = a\). These roots are significant as potential locations for the maxima or minima of the original function.

Critical points on a graph are where the derivative is zero or undefined. These points are essential as they might represent peaks, valleys, or saddle points on a graph. They are vital in analyzing and understanding the behavior of polynomial functions, particularly when looking for maximum or minimum values.
Given the equation \( f'(x) = 0 \), solving provides the critical points based on its quadratic nature:\[ x^2 - 3ax + 2a^2 = 0 \]This results in potential critical points at \(x = 2a\) and \(x = a\). Or in context, the points \(p\) and \(q\) such that

• \(p = 2a\)
• \(q = a\)
• Condition: \(p^2 = q\)

These pairs comply with the given condition \(p^2 = q\) to find the value of \(a\) that satisfies the problem: \[ (2a)^2 = a \]This leads to solving \(4a^2 = a\) and finding the correct \(a\) from given options, solidifying understanding of critical point analysis within polynomial functions.