Math Problem Statement

Укажите множество, на котором функция y = x^4/12 - x^3/3 - 9x + 1 выпукла вниз:

Solution

The image you uploaded contains a mathematical expression and four options for the domain of a function.

The function presented in the image is:

y=x412x339x+1y = \frac{x^4}{12} - \frac{x^3}{3} - 9x + 1

The question asks to identify the set on which the function is defined, meaning the domain of the function.

To determine the domain, we first observe the function is a polynomial, and polynomials are defined for all real values of xx. Therefore, the domain is (,+)(-\infty, +\infty).

Let’s look at the options:

  1. (,0)(2,+)(-\infty, 0) \cup (2, +\infty)
  2. (,+)(-\infty, +\infty)
  3. (0,2)(0, 2)
  4. (,2)(0,+)(-\infty, -2) \cup (0, +\infty)

Since the polynomial is defined for all xx, the correct answer is:

Option 2: (,+)(-\infty, +\infty).

Let me know if you need further explanations or details! Would you like more questions related to this topic?

Related questions:

  1. What are the general properties of polynomials and their domains?
  2. Can rational functions have different domains?
  3. How can you find the domain of a function with square roots or denominators?
  4. What are the conditions for a function to be undefined?
  5. How does the graph of a polynomial function behave for large values of xx?

Tip: When analyzing a function’s domain, always check for potential restrictions, such as division by zero or square roots of negative numbers, which are common causes for domain issues.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Domain of a Function

Formulas

y = x^4/12 - x^3/3 - 9x + 1

Theorems

Properties of Polynomials
Domain of a Polynomial Function

Suitable Grade Level

Grades 10-12