The question asks to estimate the area under the graph given the following table:

| x | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |

| y | 4 | 10 | 8 | 6 | 14 | 10 | 12 |

a) Use Simpson's rule with n=6 to estimate the area under the graph of a continuous function drawn through these points.

Finding the area was simple enough, after using Simpson's rule I got an estimation of 27.33333. It is the second part of the question I don't understand:

b) If it is known that -4 ≤ƒ(4)≤ 1 for all x, estimate the error involved in the approximation in part (a). Note that |Es| ≤ K(b-a)5 / 180n4 .

When finding this error bound using Simpson's rule, I would normally find the fourth derivative of the function, calculate the maximum and plug it in for K. How can I find K using only a table??

In the answer key provided it substitutes 4 for K, but I can't wrap my head around why that is. Is there a special trick to this I'm missing? I notice that when x=4, the y value is the local max of the function, is that why? But what about the fourth derivative? Help please ;_;

| x | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |

| y | 4 | 10 | 8 | 6 | 14 | 10 | 12 |

a) Use Simpson's rule with n=6 to estimate the area under the graph of a continuous function drawn through these points.

Finding the area was simple enough, after using Simpson's rule I got an estimation of 27.33333. It is the second part of the question I don't understand:

b) If it is known that -4 ≤ƒ(4)≤ 1 for all x, estimate the error involved in the approximation in part (a). Note that |Es| ≤ K(b-a)5 / 180n4 .

When finding this error bound using Simpson's rule, I would normally find the fourth derivative of the function, calculate the maximum and plug it in for K. How can I find K using only a table??

In the answer key provided it substitutes 4 for K, but I can't wrap my head around why that is. Is there a special trick to this I'm missing? I notice that when x=4, the y value is the local max of the function, is that why? But what about the fourth derivative? Help please ;_;