Difference between revisions of "Trivial Inequality"
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The trivial inequality states that <math> x^2 >= 0 </math> for all x. This is a rather useful inequality for proving that certain quantities are non-negative. The inequality appears to be obvious and unimportant, but it can be a very powerful problem solving technique. | The trivial inequality states that <math> x^2 >= 0 </math> for all x. This is a rather useful inequality for proving that certain quantities are non-negative. The inequality appears to be obvious and unimportant, but it can be a very powerful problem solving technique. | ||
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| + | === Applications === | ||
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| + | '''Maximizing and minimizing quadratic functions''' | ||
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| + | Let <math>f(x)=ax^2+bx+c</math> be a function of degree two, that is, a quadratic function, where <math>a\in\{1,-1\}</math>. If <math>{a}=1</math>, then <math>{f}</math> has only a minimum; if <math>a=-1</math>, then <math>{f}</math> has only a maximum. We can 'complete the square' in this function as follows: | ||
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| + | <math>ax^2+bx+c = a\left(x+\frac{b}{2}\right)^2-a\cdot \frac{b^2}{4}+c</math> | ||
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| + | By the trivial inequality, the minimum/maximum is then easily determined [too lazy to explain]. | ||
Revision as of 15:28, 17 June 2006
The Inequality
The trivial inequality states that 
for all x. This is a rather useful inequality for proving that certain quantities are non-negative. The inequality appears to be obvious and unimportant, but it can be a very powerful problem solving technique.
Applications
Maximizing and minimizing quadratic functions
Let 
be a function of degree two, that is, a quadratic function, where 
. If 
, then 
has only a minimum; if 
, then has only a maximum. We can 'complete the square' in this function as follows:
By the trivial inequality, the minimum/maximum is then easily determined [too lazy to explain].
