91Ó°ÊÓ

Polynomial functions are mathematical expressions involving a sum of powers of a variable, each multiplied by a coefficient. The general format for a polynomial function might look like this:

• \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)

Where:

• \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants called coefficients.
• \( n \) is a non-negative integer.

Polynomial functions are important because they are continuous and differentiable, which means they can be used to model a wide range of real-world phenomena. In the exercise, the polynomial function is given by \( T(t) = 2t^4 + t^2 - 1 \). This is a quartic polynomial, meaning the highest power of \( t \) is four.

Real-valued functions are those where the input domain contains real numbers, and the output range gives real numbers as well. They are fundamental in describing real-world situations because they can take any value on the real number line.
In practical terms, for a real-valued function, you can input any real number, and the result will always be a real number. The given function, \( T(t) = 2t^4 + t^2 - 1 \), is real-valued since it is defined and outputs real numbers for all \( t \in \mathbb{R} \).
Understanding the domain of real-valued functions, especially polynomial ones like in this exercise, is straightforward because they are typically defined for all real numbers, meaning their domain is \( (-\infty, \infty) \).

The derivative of a function represents the rate of change of the function's output with respect to its input. It tells us how fast or slow a function is changing at any given point. The derivative is often denoted as \( f'(x) \) or \( \frac{df}{dx} \).
For polynomial functions, derivatives can easily be computed using the power rule, which states that if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \). In the exercise, the derivative of \( T(t) = 2t^4 + t^2 - 1 \) is found to be \( T'(t) = 8t^3 + 2t \). This derivative will help identify critical points where the function might have a maximum, minimum, or a point of inflection.
When the derivative equals zero, these points are called critical points.

Critical points of a function are the points where the first derivative is zero or undefined. These points are significant because they can indicate where the function has a local maximum, local minimum, or a saddle point.
In the exercise, to find the critical points of \( T(t) = 2t^4 + t^2 - 1 \), we set its derivative \( T'(t) = 8t^3 + 2t \) equal to zero, getting \( t(8t^2 + 2) = 0 \). Solving this yields \( t = 0 \) as the critical point since \( 8t^2 + 2 \) does not equate to zero.
Evaluating the function at this critical point, \( T(0) = -1 \) shows that it corresponds to a local minimum of the function. This information is crucial to finding the range of the polynomial, as the range starts from this minimum value at \( -1 \) and goes to infinity, i.e., \( [ -1, \infty) \).