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September 12, 2013

Existence of Real Root

The equation x^3 + 3x - 4 has exactly one real root.


Discussion: Consider the derivative of the function f(x) = x^3 + 3x - 4 = 0 . It is 3x^2 + 3 . Note that the derivative is strictly positive ( positive times square + positive is always positive). Hence it crosses the x axis exactly once (it does cross once for large negative number will make it negative and large positive number makes it positive, hence by intermediate value property theorem it is at least once 0 in between ).

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