Let's break down the problem from the image step by step:

a. Intermediate Value Theorem (IVT) Application:

The problem asks you to apply the Intermediate Value Theorem (IVT) to show that the equation $x^3 + x + 3 = 0$ has a solution in the interval $(-2, 1)$.

The Intermediate Value Theorem states:

• If $f(x)$ is continuous on a closed interval $[a, b]$,
• And $N$ is a value strictly between $f(a)$ and $f(b)$,
• Then there is at least one $c \in (a, b)$ such that $f(c) = N$.

Here, the function is $f(x) = x^3 + x + 3$, and we are given the interval $(-2, 1)$.

Step 1: Check Continuity

The function $f(x) = x^3 + x + 3$ is a polynomial, and polynomials are continuous everywhere, so it's continuous on $(-2, 1)$. This satisfies the first condition of the IVT.

Step 2: Evaluate the Function at the Endpoints

Evaluate $f(x)$ at $x = -2$ and $x = 1$:

• $f(-2) = (-2)^3 + (-2) + 3 = -8 - 2 + 3 = -7$
• $f(1) = (1)^3 + 1 + 3 = 1 + 1 + 3 = 5$

Now we have:

• $f(-2) = -7$
• $f(1) = 5$

Step 3: Apply IVT

Since $f(-2) = -7$ and $f(1) = 5$, and 0 lies between -7 and 5, by the Intermediate Value Theorem, there exists at least one $c \in (-2, 1)$ such that $f(c) = 0$. Therefore, the equation $x^3 + x + 3 = 0$ has a solution in this interval.

b. Graphing Utility and Solutions

To find the exact solution(s), you can use a graphing calculator or utility to graph the function $f(x) = x^3 + x + 3$ and identify where the graph crosses the x-axis (i.e., where $f(x) = 0$) in the interval $(-2, 1)$.

c. Graph Illustration

Once the function is graphed, you should illustrate the curve of $x^3 + x + 3$ on the interval $(-2, 1)$, ensuring that it passes through the x-axis, confirming the existence of the solution.

Summary:

• $f(x)$ is continuous on $(-2, 1)$.
• $f(-2) = -7$, $f(1) = 5$, and 0 lies between -7 and 5.
• Therefore, by the Intermediate Value Theorem, a solution exists in the interval $(-2, 1)$.

Would you like further details on this, or help with solving any part?

Here are five related questions to expand your understanding:

1. How does the Intermediate Value Theorem work for non-polynomial functions?
2. Can there be more than one solution in the interval $(-2, 1)$?
3. What other methods can be used to solve $x^3 + x + 3 = 0$ besides the IVT?
4. How would the graph of $x^3 + x + 3$ look outside the interval $(-2, 1)$?
5. How do you verify the exact root found using graphing tools?

Tip: Always remember that polynomials are continuous everywhere, so the IVT can easily apply to any interval for polynomial functions!