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Trivial Inequality

Revision as of 12:26, 15 November 2007 by 10000th User (talk | contribs) (→‎Applications)

The Trivial Inequality states that ${x^2 \ge 0}$ for all real numbers $x$. This is a rather useful inequality for proving that certain quantities are nonnegative. The inequality appears to be obvious and unimportant, but it can be a very powerful problem solving technique.

Applications

The trivial inequality can be used to maximize and minimize quadratic functions.

After completing the square, the trivial inequality can be applied to determine the extrema of a quadratic function.

Here is an example of the important use of this inequality:

Suppose that $a,b$ are nonnegative real numbers. Starting with $(a-b)^2\geq0$, after squaring we have $a^2-2ab+b^2\geq0$. Now add $4ab$ to both sides of the inequality to get $a^2+2ab+b^2=(a+b)^2\geq4ab$. If we take the square root of both sides (since both sides are nonnegative) and divide by 2, we have the well-known Arithmetic Mean-Geometric Mean Inequality for 2 variables: $\frac{a+b}2\geq\sqrt{ab}$

Problems

Intermediate

  1. Triangle $ABC$ has $AB$$=9$ and $BC: AC=40: 41$. What is the largest area that this triangle can have?
    • Solution: First, consider the triangle in a coordinate system with vertices at $(0,0)$, $(9,0)$, and $(a,b)$.
      Applying the distance formula, we see that $\frac{ \sqrt{a^2 + b^2} }{ \sqrt{ (a-9)^2 + b^2 } } = \frac{40}{41}$.
    We want to maximize $b$, the height, with $9$ being the base. Simplifying gives $-a^2 -\frac{3200}{9}a +1600 = b^2$. To maximize $b$, we want to maximize $b^2$. So if we can write: $-(a+n)^2+m=b^2$ then $m$ is the maximum value for $b^2$. This follows directly from the trivial inequality, because if ${x^2 \ge 0}$ then plugging in $a+n$ for $x$ gives us ${(a+n)^2 \ge 0}$. So we can keep increasing the left hand side of our earlier equation until ${(a+n)^2 = 0}$. We can factor $-a^2 -\frac{3200}{9}a +1600 = b^2$ into $-(a +\frac{1600}{9})^2 +1600+(\frac{3200}{9})^2 = b^2$. We find $b$, and plug into $9\cdot\frac{1}{2} \cdot b$. Thus, the area is $9\cdot\frac{1}{2} \cdot \frac{40*41}{9} = 820$.

See also

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