91Ó°ÊÓ

Polynomials are a significant concept in mathematics, often encountered in algebra and calculus. A polynomial function is any expression involving terms with non-negative integer powers of a variable, such as \(z^3\), \(z^2\), or \(z\). They can take the form of

• Linear polynomials (e.g., \(ax + b\))
• Quadratic polynomials (e.g., \(ax^2 + bx + c\))
• Cubic polynomials (e.g., \(ax^3 + bx^2 + cx + d\))

The given function, \(f(z) = \frac{1}{3}(z^3 + z - 4)\), is a cubic polynomial. These are particularly nice to work with because they are everywhere defined, which means you can plug in any real number and expect a real output. This universality makes polynomial functions both easily applicable and robust across various mathematical contexts.

Differentiability is a property of a function that allows it to have a derivative. Put simply, if a function is differentiable at a point, it has a well-defined slope or rate of change there. For polynomial functions, like the one in our example, they are differentiable everywhere in their domain. This characteristic makes it straightforward to apply rules of differentiation to find the derivatives.For our function, we compute its derivative \(f'(z)\) by differentiating each term separately. The process involves:

• Taking the derivative of a constant times a function using the constant multiple rule.
• Applying the power rule to the terms, which states that the derivative of \(z^n\) is \(nz^{n-1}\).

Therefore, the derivative of \(f(z) = \frac{1}{3}(z^3 + z - 4)\) is:\[ f'(z) = z^2 + \frac{1}{3}. \]This differentiability on the entire real line, including on the interval \((-1, 2)\), assures that we can apply the Mean Value Theorem without issue.

Continuity is a fundamental property of functions that implies no breaks, jumps, or holes in their graphs. A function is continuous if, roughly speaking, you can draw its graph without lifting your pencil off the paper.For polynomial functions, one of their main properties is that they are continuous everywhere over real numbers. This attribute implies that within any chosen interval, like \([-1, 2]\) in our problem, the polynomial function \(f(z) = \frac{1}{3}(z^3 + z - 4)\) is smoothly connected at all points.Continuity across the specified interval is crucial in applying the Mean Value Theorem. This theorem relies on both continuity over a closed interval and differentiability over an open interval to guarantee a point where the instantaneous rate of change (derivative) equals the average rate of change.

Interval analysis is the examination of a function’s behavior over a specified range of input values. For our problem, it involves observing the polynomial \(f(z)\) on the interval \([-1, 2]\).Firstly, evaluating endpoints provides the boundary values of the function:

• \(f(-1) = -2\)
• \(f(2) = 2\)

These values help apply the Mean Value Theorem, where you determine the average change over \([-1, 2]\) and calculate where the derivative will match this rate.The practice of analyzing intervals involves setting up the Mean Value Theorem equation:\[ f'(c) = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{4}{3}. \]Then, solving:\[ z^2 + \frac{1}{3} = \frac{4}{3}, \]yields \(z = 1\) as the point \(c\) in \((-1, 2)\) where the theorem holds. This analytical approach enables us to precisely find the value where the function's behavior fits the Mean Value Theorem's requirements.