The problem asks us to determine the cubic function $f(x) = ax^3 + bx^2 + cx + d$, given the conditions:

1. $f(-3) = -100$
2. $f(-1) = 0$
3. $f(1) = 4$
4. $f(2) = 0$

We will solve this step by step.

Step 1: Write equations based on the given conditions

From $f(x) = ax^3 + bx^2 + cx + d$, substitute each condition to form equations:

1. $f(-3) = -100$: $a(-3)^3 + b(-3)^2 + c(-3) + d = -100$ $-27a + 9b - 3c + d = -100$

2. $f(-1) = 0$: $a(-1)^3 + b(-1)^2 + c(-1) + d = 0$ $-a + b - c + d = 0$

3. $f(1) = 4$: $a(1)^3 + b(1)^2 + c(1) + d = 4$ $a + b + c + d = 4$

4. $f(2) = 0$: $a(2)^3 + b(2)^2 + c(2) + d = 0$ $8a + 4b + 2c + d = 0$

Step 2: Solve the system of equations

We now have four equations:

1. $-27a + 9b - 3c + d = -100$
2. $-a + b - c + d = 0$
3. $a + b + c + d = 4$
4. $8a + 4b + 2c + d = 0$

We will solve this system for $a$, $b$, $c$, and $d$. Let me calculate. sympy import symbols, Eq, solve

Define variables

a, b, c, d = symbols('a b c d')

Define the equations

eq1 = Eq(-27a + 9b - 3c + d, -100) eq2 = Eq(-a + b - c + d, 0) eq3 = Eq(a + b + c + d, 4) eq4 = Eq(8a + 4b + 2c + d, 0)

Solve the system of equations

solution = solve((eq1, eq2, eq3, eq4), (a, b, c, d)) solutionThe solution to the system of equations is:

$a = 2, \quad b = -6, \quad c = 0, \quad d = 8$

Thus, the cubic function is:

$f(x) = 2x^3 - 6x^2 + 8$

Would you like me to verify or explain any part of this solution further?

Related Questions:

1. What does each term in a cubic equation represent in terms of graph behavior?
2. How can we test if this function satisfies the given conditions?
3. What are the critical points of the function $f(x)$?
4. How would the function change if the condition $f(2) = 0$ was modified?
5. How do you convert a cubic function into its factored form?

Tip:

To check if your solution is correct, substitute all given conditions into your final function and verify if they hold true.