# Thread: Intermediate Value Theorem

1. ## Intermediate Value Theorem

Can somebody help me with this problem. I'm trying to just understand it but having a hard time. Thanks

Use the Intermediate Value Theorem to show that there is a root of the equation x3+2x2=42 on the interval (0, 3). Your final answer should state your chosen values of N, f(x), a, and b.

2. IVT: If f is continuous on the interval $\displaystyle [a,b]$ and $\displaystyle s$ is a number between $\displaystyle f(a)$ and $\displaystyle f(b)$, then there exists a number $\displaystyle c$ in the interval $\displaystyle [a,b]$ such that $\displaystyle f(c) = s$

For your question, imagine s = 0 (i.e. a root)

Let $\displaystyle f(x) = x^3 + 2x^2 - 42$. Note: $\displaystyle f(0) = -42$ and $\displaystyle f(3) = 444$. So what can you conclude from IVT?

3. So anything in between these two points 0 and 3 on the X-axis which will be C, will equal automatically equal S, which is between F(a) and F(b)?

4. The theorem says that if you give me a number between f(0) and f(3), call it s, I can guarantee you that I will find you a number c in between 0 and 3 such that f(c) = s.

Put s = 0 and you're done.