Question

The function f is continuous on the closed interval [2, 4] and twice differentiable on the open interval (2, 4). If f'(3) = 2 and f"(3) < 0 on the open interval (2, 4), which could be a table of values for f?

Answer 1

A table of values for the function f should show an increasing trend around x=3 but with the function slowing down or reaching a peak at x=3, according to the information that f'(3) = 2 and f"(3) < 0.

Step-by-step explanation:

The given information indicates that function f is continuous on the interval [2, 4] and has a derivative f'(3) = 2, which means the slope of the tangent line to the graph of the function at x=3 is positive, indicating an increasing function at that point. Additionally, f"(3) < 0 tells us that the concavity of the function is downward at x=3, which means the function is slowing down as it increases at x=3 or has reached a local maximum. A table of values for f should therefore show increasing values leading up to x=3, and if x>3, the values should start to either decrease or increase at a decreasing rate. However, without the actual function, this description cannot be translated to a specific table of values, but the understanding of the derivative information can help infer the shape of the graph around x=3.

Answer 2

A possible table of values for f could be x = 2, f(x) = 4; x = 3, f(x) = 6; x = 4, f(x) = 7.

Step-by-step explanation:

To find a possible table of values for f, we need to consider the given conditions.

We know that f'(3) = 2, which tells us the slope of the tangent line to the graph of f at x = 3 is 2.

Since f is continuous on [2, 4], this means that as x approaches 3 from both sides, the graph of f becomes steeper at x = 3 from the left side to the right side.

Additionally, f''(3) < 0 means that the concavity of f at x = 3 is downward. Based on these conditions, a possible table of values for f could be:

2.0 a

2.5 b

3.0 c

3.5 d

4.0 e