How to Determine Constants of a Cubic Function for Inflection and Maximum Points

Understanding how to determine the constants a, b, c, and d for a cubic function, so that it has a specific inflection point and a local maximum, is a fundamental skill in calculus and can be applied in various fields such as engineering, physics, and economics. This guide will walk you through the process using detailed steps and examples.

Step 1: Identifying the Point of Inflection

Consider the general form of a cubic function:

Step 1.1: Finding the First and Second Derivatives

The first and second derivatives of the function fx ax^3 bx^2 cx d are:

First Derivative: fx' 3ax^2 2bx c

Second Derivative: fx'' 6ax 2b

Step 1.2: Determining the Point of Inflection

A point of inflection occurs where the second derivative changes sign, which is determined by setting the second derivative equal to zero:

fx'' 6ax 2b 0

At the origin, x 0 and fx''(0) 2b. For a point of inflection at the origin:

2b 0

Conclusion: b 0

Step 2: Identifying the Local Maximum

A local maximum occurs where the first derivative is zero, and the function value at that point is a specific value. Using the relationship derived from the point of inflection, we can proceed to the local maximum condition.

Step 2.1: Simplifying the Function

With b 0, the function simplifies to:

fx ax^3 cx d

Step 2.2: Setting the First Derivative to Zero

The first derivative now is:

fx' 3ax^2 c

Setting fx'(-2) 0 to find the local maximum at x -2:

3a(-2)^2 c 0

12a c 0

Conclusion: c -12a

Step 2.3: Ensuring the Function Value at the Point

Given that the function value at x -2 is 4:

fx(-2) a(-2)^3 c(-2) d 4

-8a - 2c d 4

Substituting c -12a:

-8a - 2(-12a) d 4

-8a 24a d 4

16a d 4

Conclusion: d 4 - 16a

Step 3: Summary of Relationships

From the above steps, we have derived the following relationships:

b 0 c -12a d 4 - 16a

Step 4: Choosing a Value for a

To express the function in a specific form, we can choose a value for a. Let's select a 1 for simplicity:

a 1 b 0 c -12(1) -12 d 4 - 16(1) -12

The final function is:

fx x^3 - 12x - 12

Conclusion

The constants for the function fx are:

a 1 b 0 c -12 d -12

Thus, the function is:

fx x^3 - 12x - 12