# Differentiation : Existence of Solutions

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I want to prove, for the numbers a and b, that the following equation has exactly three solutions if and only if 4a^3 + 27b^2 < 0:

x^3 + ax + b = 0,

x in R

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#### Solution Preview

Because, f(x) = x^3 + ax + b

=> f'(x) = 3x^2 + a

for maxima-minima:

f'(x) = 0 => x = +sqrt(-a/3) or x = -sqrt(-a/3)

f''(x) = 6x

f''(x=+sqrt(-a/3)) = ...

#### Solution Summary

It is proved that there are a set number of solutions for a particular function. Maxima and minima's are analyzed for functions.

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