91Ó°ÊÓ

In calculus, a **derivative** is a measure that shows how a function changes as its input changes. It represents the rate of change or the slope of the tangent line to the curve of a function at any given point. For polynomial functions like \( f(x) = x^4 + 4x^3 - 8x^2 + 64 \), finding the derivative involves using basic differentiation rules.

Key rules to remember include:

• The **power rule**: For any term \( ax^n \), the derivative is \( anx^{n-1} \). For example, \( \frac{d}{dx}(x^4) = 4x^3 \).
• The derivative of a **constant** is zero. For instance, \( \frac{d}{dx}(64) = 0 \).
• The derivative is computed term by term if the function is a sum or difference of terms.

After finding the derivatives of all parts, they are combined to form the derivative of the whole function. In our exercise, the derivative \( f'(x) = 4x^3 + 12x^2 - 16x \) was essential to find critical numbers. These occur where the function tends to be at its lowest or highest points, called **extrema** (minima and maxima).

Polynomial functions are expressions consisting of variables raised to various exponents and multiplied by coefficients. They can include constant terms too. An example of a polynomial function is \( f(x) = x^4 + 4x^3 - 8x^2 + 64 \). Polynomial functions are known for their **smooth and continuous graphs** which are defined everywhere so they don’t have breaks or jumps.

Key characteristics of polynomial functions include:

• **Degree**: This is the highest power of the variable. Here, it's 4, making it a quartic polynomial.
• **Coefficients**: The numbers before each variable, like 4 in \( 4x^3 \).
• **Behavior at infinity**: Polynomials behave predictably at large values of \( x \), either going to positive or negative infinity depending on the leading term.

Understanding polynomials is crucial because they lay the groundwork for finding derivatives, like in our task. These derivatives help identify critical numbers, giving insight into the function's maximum and minimum values, where the function reverses direction or flattens out.

The **quadratic formula** is a versatile mathematical tool used to find solutions to quadratic equations, which are equations in the form \( ax^2 + bx + c = 0 \). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \( a \), \( b \), and \( c \) are coefficients from the equation.
When used in the exercise, the quadratic formula helped solve the quadratic factor \( x^2 + 3x - 4 = 0 \) after factoring the derivative of the polynomial function. Here's a breakdown of its application:

• Calculate the **discriminant** \( b^2 - 4ac \), which determines the nature of solutions (real and distinct, real and equal, or complex).
• Utilize the **plus-minus sign (±)** to account for both possible outcomes from taking a square root, leading to two potential solutions for \( x \).

The formula is powerful because it provides a definitive method for finding roots of any quadratic expression. In our problem, it resulted in the critical numbers \( x = 1 \) and \( x = -4 \), key points that tell us important features of the function's graph.