Signs of Products, and the Trivial Inequality (x² ≥ 0)
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- Inequalities (Part 1) Main Inequality For all real numbers xxx, x2≥0x^2 \ge 0x2≥0, with equality if and only if x=0x = 0x=0. This is called the main inequality, and is the inequ…
By Zi Song Yeoh, Alex Federspiel
- Inequalities (Part 6) This is not really a post on applications of famous inequalities to solve problems. This post will be on how to make an inequality problem (that might or m…
By Zi Song Yeoh
- The Trivial Inequality The trivial inequality refers to the well-known fact that the square of a real number will always be greater than or equal to 0. You probably realized thi…
By Yujin Kim
- We'll consider 3 situations: When x>0x>0x>0: If x>0x>0x>0, we have x2≥0x^2\ge0x2≥0 for all real values of xxx. When x=0x=0x=0: Here, of course, x2=0x^2=0x2=0. When x<0x<0x<0: If…
By Atrayee Samanta, Victor Wang
Problems 5
- For what integer values of nnn is xn+x4+x2x^n + x^4 + x^2xn+x4+x2 nonnegative ∀x∈R\forall x \in \mathbb{R}∀x∈R?
- Let aaa be a nonzero real constant. Given that xxx is a real number, find the maximum value of: ∣x∣x2+a2\frac{|x|}{x^2+a^2}x2+a2∣x∣ as well as the value of xxx at which this ma…
- Consider a rectangle with perimeter PPP. In terms of PPP, find the maximum possible area of the rectangle, and describe the rectangle that has this optimal area.
- Find the solution to the equation (x−2)2+(y−3)2+(z−4)2=0(x-2)^2+(y-3)^2+(z-4)^2=0(x−2)2+(y−3)2+(z−4)2=0.
- If reals a,b,ca,b,ca,b,c satisfy a2+b2+c2=ab+bc+ac,a^2+b^2+c^2=ab+bc+ac,a2+b2+c2=ab+bc+ac, how many possible values are there for a−b?a-b?a−b?