The question discusses the Intermediate Value Theorem (IVT) and asks for two sketches based on a function $f$ defined on the interval $[1, 4]$, with some discontinuity at $x = 3$. Here's the analysis:

Intermediate Value Theorem (IVT):

The IVT states that if a function is continuous on a closed interval $[a, b]$, and $N$ is any value between $f(a)$ and $f(b)$, then there exists at least one point $c \in (a, b)$ such that $f(c) = N$.

In this case:

• $f(1) = 1$
• $f(4) = 3$
• There is a discontinuity at $x = 3$.
• You are given $N = 2$ and need to explore whether the function $f$ might or might not satisfy the conclusion of the IVT, given that it's discontinuous at $x = 3$.

Two scenarios:

1. Graph where the function does not satisfy the conclusion of the IVT:

   • The function could "jump" at $x = 3$, meaning there's a break or gap at that point.
   • Since the function is discontinuous at $x = 3$, it's possible that the function never attains the value $f(x) = 2$ in the interval $[1, 4]$.
   • In this case, the discontinuity prevents the IVT from applying.

2. Graph where the function might still satisfy the conclusion of the IVT:

   • Even though there is a discontinuity at $x = 3$, the function could still take the value $2$ at some point between $1$ and $4$, perhaps at $x = 2$.
   • The function could be continuous on $[1, 3)$ and $(3, 4]$, with a discontinuity at $x = 3$, but still pass through the value $2$ at some point, which would satisfy the conclusion of the IVT.

Would you like me to create the sketches for both cases or explain the details further?

Related Questions:

1. What are the key conditions for the Intermediate Value Theorem to apply?
2. Why is the discontinuity at $x = 3$ significant for the IVT?
3. Could there be other values for $N$ that the function $f$ must satisfy under the IVT?
4. What happens to the IVT if a function is only piecewise continuous?
5. How does the behavior of $f(x)$ at $x = 3$ affect the possibility of satisfying the IVT?

Tip:

When verifying the IVT, always check for both the continuity condition and the value condition in the interval.