Algebra
Polynomial Inequalities
If \(x+y=20\), what is the maximum value of \(2xy\)?
A rectangular wall for a chicken coop is to be made out of a \( 44 \mbox{ m}\) wire mesh. Let \(x\) be the shorter side of the rectangle in meters. If the area of the rectangle, \(A\), is \( 120 \mbox{ m}^2 < A < 121 \mbox{ m}^2\), then \(x\) must satisfy \( a < x < b \). What is the value of \(a+b\)?
If \( x\) and \(y \) are positive numbers, what is the minimum value of \( (3x + 4y)\left(\frac{27}{x} + \frac{4}{y} \right) \)?
Hint: Let \( x = 2 ^ {2015} \).
What is the maximum value of \(x\) that satisfies the inequality \( (x - 6)(x^2 - 4x + 9) \leq 0 \)?