Purchase Solution

# Differentiation : Existence of Solutions

Not what you're looking for?

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

##### 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.

##### 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 provided by:
###### Education
• BEng, Allahabad University, India
• MSc , Pune University, India
• PhD (IP), Pune University, India
###### Recent Feedback
• " In question 2, you incorrectly add in the \$3.00 dividend that was just paid to determine the value of the stock price using the dividend discount model. In question 4 response, it should have also been recognized that dividend discount models are not useful if any of the parameters used in the model are inaccurate. "
• "feedback: fail to recognize the operating cash flow will not begin until the end of year 3."
• "Answer was correct"
• "Great thanks"
• "Perfect solution..thank you"

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.