To solve for the values of $k$ such that the equation $f(x) = 2$ has at least two solutions in the interval $[-1, 1]$, let's analyze the problem step by step.

Step 1: Understanding the problem

We are given that the function $f(x)$ is continuous on the interval $[-1, 1]$, and the values of $f(x)$ are provided at three points:

• $f(-1) = 3$
• $f(0) = k$
• $f(1) = 5$

We are tasked with finding the values of $k$ for which the equation $f(x) = 2$ has at least two solutions in the interval $[-1, 1]$.

Step 2: Applying the Intermediate Value Theorem

Since $f(x)$ is continuous, we can apply the Intermediate Value Theorem (IVT). This theorem states that if a function is continuous on a closed interval $[a, b]$ and takes values $f(a)$ and $f(b)$ at the endpoints, then for any value $y$ between $f(a)$ and $f(b)$, there exists at least one $c \in (a, b)$ such that $f(c) = y$.

We are looking for the values of $k$ where $f(x) = 2$ has at least two solutions. Let's examine the intervals $[-1, 0]$ and $[0, 1]$ separately.

Step 3: Finding solutions in the interval $[-1, 0]$

• On the interval $[-1, 0]$, we know that $f(-1) = 3$ and $f(0) = k$.
• For the equation $f(x) = 2$ to have a solution in this interval, $2$ must lie between $f(-1) = 3$ and $f(0) = k$.
• This gives the condition: $3 \geq 2 \geq k$ or equivalently, $k \leq 2$.

Step 4: Finding solutions in the interval $[0, 1]$

• On the interval $[0, 1]$, we know that $f(0) = k$ and $f(1) = 5$.
• For the equation $f(x) = 2$ to have a solution in this interval, $2$ must lie between $f(0) = k$ and $f(1) = 5$.
• This gives the condition: $k \leq 2 \leq 5$ or equivalently, $k \leq 2$.

Step 5: Combining the conditions

From both intervals, we obtained the condition $k \leq 2$. This ensures that there is at least one solution in each interval, leading to two solutions in total if $k < 2$.

Final Answer:

For the equation $f(x) = 2$ to have at least two solutions, $k$ must satisfy: $k < 2$ If $k = 2$, there would be exactly one solution at $x = 0$.

---

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How does the Intermediate Value Theorem apply to functions that are not continuous?
2. What happens to the number of solutions if $k > 2$?
3. Can you explain the difference between "at least two solutions" and "exactly two solutions" in this context?
4. How would the solution change if the function was not continuous on $[-1, 1]$?
5. What other types of problems can be solved using the Intermediate Value Theorem?

Tip: The Intermediate Value Theorem is a powerful tool for proving the existence of solutions but does not guarantee how many solutions exist. Always check boundary conditions carefully.