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Polynomial Inequalities

When does revenue exceed cost? When will one race car pass another? Model these values with polynomials and use polynomial inequalities to solve questions like these and more.

Problem Solving

         

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 \( (3x + 4y)\left(\frac{27}{x} + \frac{4}{y} \right) \)?

Which is larger?

Hint: Let \( x = 2 ^ {2015} \).

What is the maximum value of \(x\) that satisfies the inequality \( (x - 6)(x^2 - 4x + 9) \leq 0 \)?

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