Use cubic regression to find a function that fits the following points. $$ \begin{array}{l}(-2,-16)(1,5)(3,59)(6,440) \ {[?] x^{3}+\square x+\square}\end{array} $$

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To find a cubic function that exactly fits the given points $$(-2, -16)$$, $$(1, 5)$$, $$(3, 59)$$, and $$(6, 440)$$, we'll assume the general form of a cubic polynomial:

$$
f(x) = ax^3 + bx^2 + cx + d
$$

### Step 1: Set Up the System of Equations

Plugging each point into the equation gives us four equations:

1. For $$(-2, -16)$$:
   $$
   a(-2)^3 + b(-2)^2 + c(-2) + d = -16 \quad \Rightarrow \quad -8a + 4b - 2c + d = -16
   $$
   
2. For $$(1, 5)$$:
   $$
   a(1)^3 + b(1)^2 + c(1) + d = 5 \quad \Rightarrow \quad a + b + c + d = 5
   $$
   
3. For $$(3, 59)$$:
   $$
   a(3)^3 + b(3)^2 + c(3) + d = 59 \quad \Rightarrow \quad 27a + 9b + 3c + d = 59
   $$
   
4. For $$(6, 440)$$:
   $$
   a(6)^3 + b(6)^2 + c(6) + d = 440 \quad \Rightarrow \quad 216a + 36b + 6c + d = 440
   $$

### Step 2: Solve the System

Subtract the second equation from the first, third, and fourth to eliminate $$d$$:

1. $$ -9a + 3b - 3c = -21 $$ \quad (Simplified to $$3a - b + c = 7$$)
2. $$ 26a + 8b + 2c = 54 $$ \quad (Simplified to $$2a + b = 4$$)
3. $$ 215a + 35b + 5c = 435 $$ \quad (Simplified to $$5a + b = 10$$)

From the simplified equations:

- Subtract $$2a + b = 4$$ from $$5a + b = 10$$ to find $$a = 2$$.
- Substitute $$a = 2$$ into $$2a + b = 4$$ to find $$b = 0$$.
- Substitute $$a = 2$$ and $$b = 0$$ into $$3a - b + c = 7$$ to find $$c = 1$$.
- Finally, substitute $$a = 2$$, $$b = 0$$, and $$c = 1$$ into $$a + b + c + d = 5$$ to find $$d = 2$$.

### Step 3: Write the Final Function

Substituting $$a = 2$$, $$b = 0$$, $$c = 1$$, and $$d = 2$$ into the general form gives:

$$
f(x) = 2x^3 + x + 2
$$

### Verification

Let's verify the function with the given points:

- $$ f(-2) = 2(-2)^3 + (-2) + 2 = -16 - 2 + 2 = -16 $$
- $$ f(1) = 2(1)^3 + 1 + 2 = 5 $$
- $$ f(3) = 2(3)^3 + 3 + 2 = 54 + 3 + 2 = 59 $$
- $$ f(6) = 2(6)^3 + 6 + 2 = 432 + 6 + 2 = 440 $$

All points satisfy the equation, confirming that the function is correct.

**Final Answer:**

$$
f(x) = 2\,x^{3} + x + 2
$$
The cubic function that fits the points is $$ f(x) = 2x^{3} + x + 2 $$.

Use cubic regression to find a function that fits the following points. \( \begin{array}{l}(-2,-16)(1,5)(3,59)(6,440) \\ {[?] x^{3}+\square x+\square}\end{array} \)

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