Math Problem Statement
Let S be the statement ∃M > 0, ∀x ∈ R, (x(4 − x) < M ) That is, there is a real number M > 0 such that for all real numbers x, x(4 − x) < M . If S is true, prove it. If S is false, disprove it by stating and proving its negation
Solution
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Inequalities
Quadratic Functions
Real Analysis
Formulas
f(x) = x(4 - x) = 4x - x^2
Vertex formula: x = -b / 2a for quadratic functions ax^2 + bx + c
Theorems
Quadratic Maximum Theorem: For a quadratic function with negative leading coefficient, the maximum value occurs at the vertex.
Suitable Grade Level
Undergraduate
Related Recommendation
Understanding the Function x(1 - x) and Inequalities Between Real Numbers
Proving the Inequality x(1-x) ≤ 1/4 for x in [0,1]
Function Analysis of x^4, |x| - 3, √(x^2 + 2), and 1/x^2
Detailed Analysis of f(x) = (x-1)^2 - 4: Function Definition, Graph, and Monotonicity
Maximum Value of a Quadratic Function: f(x) = -3(x - 2)² - 4