Final answer:
The question asks to determine whether certain functions are 'o(x²)', which implies a comparison of their growth rates to the square function.
Step-by-step explanation:
The question pertains to the classification of functions according to their growth rates, particularly in comparison to the function g(x) = x², and utilizes the Big O notation to describe upper bounds on the growth rate of these functions. The notation 'o(x²)' is used in mathematics to denote a function that grows strictly slower than x² as x approaches infinity. To determine whether a function is 'o(x²)', we would examine the limit of the ratio of the function to x² as x approaches infinity. If this limit is zero, then the function is considered 'o(x²)'.
We can directly apply the properties of even and odd functions to assist in analyzing the given functions. An even function is symmetric about the y-axis, meaning that f(x) = f(-x), while an odd function is symmetric about the origin, implying that -f(x) = f(-x). A product of two even functions is even, and a product of two odd functions is also even. Therefore, a function such as x²e-x² (even×even) or x sin x (odd×odd) would both be classified as even, hence have symmetry about the y-axis and possibly grow comparably to x², depending on additional context provided by the function. However, a product of an odd and an even function, such as xe-x², is odd.
Understanding the integral of odd and even functions is crucial. The integral of an odd function over a symmetric interval centered at the origin is zero because the positive and negative areas cancel each other out. This property is useful for evaluating the integrals of functions that are products of even and odd functions over symmetric intervals.
In the context of quadratic equations or second-order polynomials, 'o(x²)' becomes particularly relevant as these are functions that involve terms up to x². For instance, functions like these are often encountered when dealing with quadratic equations, which are of the form ax² + bx + c = 0, and can be solved using methods like completing the square, factoring, or the quadratic formula.