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When you study calculus, you will encounter the concept of **critical points**. Critical points are the values of a variable, usually denoted as \(x\), where a function's derivative\(y'\) is either zero or undefined. These points are significant because they help identify where a function might change from increasing to decreasing or vice versa.
For a given polynomial like \(y=x^4-10x^2+9\), critical points are determined by setting the derivative \(y'\) to zero and solving for \(x\). These solutions are potential turning points of the function.
In our specific example, the derivative \(y'=4x^3-20x\) equals zero at \(x=-\sqrt{5}, 0, \sqrt{5}\). Thus, these are the critical points which need further examination to determine the nature of the function at these points.

• Critical points help identify local maxima and minima.
• They are essential in evaluating where the function changes its behavior.

The **derivative** is a fundamental concept in calculus. It measures how a function changes as its input changes. Simply put, the derivative tells you the slope of the function at any given point.
For polynomial functions, finding the derivative is straightforward. You apply the power rule, which says the derivative of \(x^n\) is \(nx^{n-1}\).
In our problem, we are given \(y=x^4-10x^2+9\). By applying the power rule, the derivative becomes \(y'=4x^3-20x\).
Derivatives provide critical information about a function’s behavior such as:

• Telling us where the function is increasing or decreasing.
• Helping locate critical points by setting the derivative to zero.
• Determining the concavity of the function.

Understanding derivatives is essential for analyzing and sketching the curves of functions.

A **polynomial function** is a mathematical expression involving a sum of powers of one or more variables multiplied by coefficients. They have degrees, determined by the largest exponent in the function.
Our example, \(y=x^4-10x^2+9\), is a fourth-degree polynomial because the largest power of \(x\) is four. Characteristics of polynomial functions include being continuous and smooth, meaning they have no gaps or sharp corners.
Polynomials often exhibit several critical points and change direction several times based on their degree. Analyzing a polynomial involves:

• Finding derivatives to locate critical points.
• Applying the first derivative test to determine increasing or decreasing intervals.
• Utilizing these intervals to sketch the polynomial’s graph.

Polynomial functions play an essential role in calculus and many real-world applications.

When examining a function's graph, you often look at where the function is **increasing** and **decreasing**. An interval is increasing if the function values increase as \(x\) moves from left to right, and decreasing if those values decrease.
We use the derivative \(y'\) to identify these intervals. If \(y' > 0\), the function is increasing in that interval. If \(y' < 0\), the function is decreasing.
For our function \(y=x^4-10x^2+9\), the derivative \(y'=4x^3-20x\) helps us find:

• Increasing intervals: \((-fty, -\sqrt{5})\) and \((0, \sqrt{5})\).
• Decreasing intervals: \((-\sqrt{5}, 0)\) and \((\sqrt{5}, fty)\).

These intervals help in sketching the curve and understanding the nature of the function’s peaks and troughs. Understanding these regions is key to mastering calculus and graph interpretations.