Calculus, derivatives (curve sketching 2)

In summary, the conversation revolves around finding the values of a and b in the function f(x)=x3+a2+bx which has a local minimum value at \frac{-2}{9}\sqrt{3}. The individual is seeking help in determining the correct derivative of the function and is given a hint to use their understanding of cubic equations. The key to solving the problem is identifying the relationship between the roots of f(x) and f'(x).
  • #1
physics604
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2
1. If the function f(x)=x3+a2+bx has the local minimum value at [itex]\frac{-2}{9}[/itex][itex]\sqrt{3}[/itex], what are the values of
and a and b?


Homework Equations

$$f'(x)=0$$

The Attempt at a Solution



I automatically took the derivative, getting $$f'(x)=3x^2+2ax+b$$ However, I have no idea where to go from here because I only know one root ([itex]\frac{-2}{9}[/itex][itex]\sqrt{3}[/itex]) and not the other. Can someone give me a hint?
 
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  • #2
Your derivative does not follow from the given function.
$$f(x)=x^3+a^2+bx \implies f'(x)=3x^2+b$$ Your derivative is of $$f(x)=x^3+ax^2+bx+c$$ ... which is correct?

(I'm kinda leaning towards the second one with c=0 but I'd like to be sure.)

You are supposed to use your understanding of cubic equations to help you, not just algebra.
I suspect that has been the problem in both your questions I've seen so far.
 
  • #3
Do you know any relationships between the inflexion point of a cubic with that of its turning points? If not, use a graphing calculator to sketch a few cubics that have distinct local min and max points, and see if you can notice anything between those and the inflexion point. Maybe try finding the inflexion point in each example to make it more obvious.
 
  • #4
Lynchpin: roots of f(x) and f'(x).
(Assuming my suspicion is correct.)
 

1. What is calculus and why is it important?

Calculus is a branch of mathematics that deals with change and motion. It is important because it is used to model and analyze a wide range of natural phenomena, making it a fundamental tool in many scientific and engineering fields.

2. What are derivatives and how are they used in calculus?

Derivatives are a mathematical concept that represents the instantaneous rate of change of a function. In calculus, derivatives are used to find the slope of a curve at a specific point, which can help in determining critical points, optimization, and curve sketching.

3. What is curve sketching and why is it useful?

Curve sketching is the process of graphically representing the behavior of a function. It involves analyzing the key features of a function, such as its domain, range, intercepts, and asymptotes, in order to create an accurate graph. This is useful in understanding and visualizing the behavior of a function.

4. How do I find the critical points of a function?

The critical points of a function are the points where the derivative of the function is equal to zero or undefined. To find these points, you can take the derivative of the function and set it equal to zero, then solve for the variable. The resulting values will be the critical points of the function.

5. What is the process for finding the concavity of a curve?

The concavity of a curve is determined by the second derivative of the function. If the second derivative is positive, the curve is concave up, and if it is negative, the curve is concave down. To find the concavity of a curve, take the second derivative and analyze its sign throughout the domain of the function.

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