Suppose $f\left(x\right)$ is continuous on the interval $[0,4]$ and $f\left(x\right)>0$ for all $x$ in $(1,3)$. Which of the following could be the entire interval over which $f\left(x\right)$ is positive?

• A. $(0,4)$
• B. $[0,4]$
• C. $(1,3)$
• D. $[1,3]$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem statement. The function f(x) is continuous on the interval [0, 4], meaning there are no breaks or gaps in the graph of f(x) within this interval. Additionally, f(x) > 0 for all x in the open interval (1, 3), meaning the function is strictly positive between x = 1 and x = 3.
2. Step 2: Analyze the intervals provided in the options. The interval (0, 4) includes all values between 0 and 4 but excludes the endpoints. The interval [0, 4] includes all values between 0 and 4, including the endpoints. The interval (1, 3) includes all values strictly between 1 and 3, excluding the endpoints. The interval [1, 3] includes all values between 1 and 3, including the endpoints.
3. Step 3: Consider the condition f(x) > 0 for all x in (1, 3). This implies that the function is positive only within the open interval (1, 3). Outside this interval, the function could potentially be zero or negative, but we are not given explicit information about f(x) outside (1, 3).
4. Step 4: Evaluate the options based on the given information. The interval (0, 4) and [0, 4] cannot be correct because we are not guaranteed that f(x) > 0 outside (1, 3). The interval [1, 3] includes the endpoints 1 and 3, but the problem specifies that f(x) > 0 only for x in (1, 3), excluding the endpoints.
5. Step 5: Conclude that the correct interval over which f(x) is positive is (1, 3), as this matches the condition f(x) > 0 for all x in (1, 3) and excludes the endpoints where positivity is not guaranteed.