Finding Unknown Constants for a Cubic Function with Given Derivatives

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Homework Help Overview

The discussion revolves around determining the constants a, b, c, and d in the cubic function f(x) = ax^3 + bx^2 + cx + d, given specific conditions on its first and second derivatives at certain points.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants explore the implications of the function's derivatives at specific points, questioning the continuity and behavior of the function. There is a focus on identifying additional equations needed to solve for the unknown constants, with some participants discussing the meaning of x-intercepts and function values at specific points.

Discussion Status

The conversation is actively exploring the relationships between the function and its derivatives. Some participants have identified two equations based on the given conditions, while others are prompting for further clarification and additional equations needed to fully resolve the problem.

Contextual Notes

There is a noted confusion regarding the definitions of x-intercepts and function values at specific points, which may affect the clarity of the discussion. The original poster has not provided all necessary equations to solve for the unknowns.

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Homework Statement


determine the constants a,b,c, and d so that the function f(x)=ax^3+bx^2+cx+d has its first derivative equal to 4 at the point (1,0) and its second derivative equal to 5 at the point (2,4)

Homework Equations

The Attempt at a Solution


I found the first and second derivative
f'(x)=3ax^2+2bx+c
f''(x)=6ax+2b

I set
5=12a+2b
I also set
4=3a+2b+c
I get stuck trying to find the two different variables when working with the second derivative first. I know I have to substitute for the unknowns, but I don't know where to start.
 
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Tricky problem but if a function f(x) has a derivative at some specific point then what can you say about f(x)?
 
That f(x) is continuous at that specific point?
 
Yes, that's true, but I am thinking even more obviously than that?
 
You tell me.
 
Oh, it's an x intercept.
 
No, it's not.

f(1)=?

You have 4 unknowns so you need 4 equations to solve this. You came up with two of them. What are the other two? It's pretty obvious.
 
SammyS said:
What is f(1) ?
MrJamesta said:
Oh, it's an x intercept.

paisiello2 said:
No, it's not.
Just to be clear, and to reduce confusion on the part of the OP, 1 is an x-intercept.
 
  • #10
Well, what is an x-intercept? I think it is the value of the function when x=0. But f(1) is the value of the function when x=1. So it is not an x-intercept.

Regardless, it is irrelevant to solve the problem. What are the other two equations?
 
  • #11
paisiello2 said:
Well, what is an x-intercept? I think it is the value of the function when x=0.
No, that's the y-intercept, a point on the y-axis.
paisiello2 said:
But f(1) is the value of the function when x=1. So it is not an x-intercept.

Regardless, it is irrelevant to solve the problem.
But your reply to the OP was incorrect, possibly steering him/her in the wrong direction.
 
  • #12
To get back to the original question:
MrJamesta said:
Oh, it's an x intercept.
Right. Can you use it to find another equation?
 
  • #13
I found
0=a+b+c+d
4=8a+4b+2c+d
 
  • #14
MrJamesta said:
I found
0=a+b+c+d
4=8a+4b+2c+d
Now, use the other two equations and solve for the unknown coefficients.
 

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