Math Problem Statement

حل الاسئلة

حل مسائل التفاضل والتكامل ودراسة اشارات الدوال والجدول البياني لها.

Solution

The image contains handwritten mathematical problems in Arabic. Here's a brief outline of the questions and how to approach solving them:

Problem 1:

The first part introduces a function f(x)=2x4+x3+1f(x) = -2x^4 + x^3 + 1 and asks to:

  1. Study the sign of the function ff — meaning to analyze where f(x)f(x) is positive, negative, or zero.
  2. Construct a variation table for ff.
  3. Calculate f(1)f(1).
  4. Determine the function's domain and behavior for specific intervals.

Solution for Problem 1:

To address this:

  • Domain: Since it's a polynomial, the domain is R\mathbb{R}.
  • Sign study: Solve f(x)=0f(x) = 0 to find critical points. Factor or apply methods to solve higher-degree polynomials.
  • Variation table: Derive f(x)f'(x) to find the intervals of increase or decrease.
  • f(1)f(1): Direct substitution gives f(1)=2(1)4+(1)3+1=0f(1) = -2(1)^4 + (1)^3 + 1 = 0.

Problem 2:

The second problem asks for the function g(x)=x12x+12g(x) = \sqrt{x} - \frac{1}{2x} + \frac{1}{2}.

  1. Study the function's sign and variation.
  2. Determine points of interest and its behavior near zero and infinity.

Solution for Problem 2:

  • Domain: The square root function restricts the domain to x0x \geq 0, excluding x=0x = 0 due to division by zero.
  • Sign study: Analyze the function’s behavior by solving g(x)=0g(x) = 0.
  • Asymptotes: Analyze the limits as x0+x \to 0^+ and x+x \to +\infty.

There are additional steps in the image, like forming tables and studying other properties.

Would you like me to solve one of these problems in more detail?

5 Questions to Expand:

  1. How do you solve higher-degree polynomial equations like f(x)=0f(x) = 0?
  2. What is the role of a variation table in function analysis?
  3. How can we determine limits at infinity for functions like g(x)g(x)?
  4. What methods are used to find critical points of a function?
  5. How do you graph functions with complex terms like square roots?

Tip: Always begin by determining the domain of a function to understand any restrictions before proceeding with analysis.

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Math Problem Analysis

Mathematical Concepts

Function Analysis
Polynomials
Square Root Functions
Differentiation
Limits

Formulas

f(x) = -2x^4 + x^3 + 1
g(x) = √x - 1/2x + 1/2
Derivative formula f'(x)
Limit as x approaches 0 and ∞

Theorems

Sign Study
Limit Theorems
Critical Points and Asymptotes
Variation Table Construction

Suitable Grade Level

University level (Calculus and Function Analysis)