x & 0 & 1 & 2 & 3 & 4 & 5 f(x) & 80 & 77 & 68 & 53 & 32 & 5 The table above gives selected values for a continuous function f . If f is decreasing over the closed interval [0,5] , which of the following could be the value of int_(0)^5 f(x) d x ? A. 235 B. 275 C. 310 D. 400

To determine which value could be the integral \(\int_{0}^{5} f(x) \, dx\), we need to consider that \(f(x)\) is a decreasing function over the interval \([0, 5]\). This means that the function starts at a higher value and decreases as \(x\) increases from 0 to 5.<br /><br />The integral \(\int_{0}^{5} f(x) \, dx\) represents the area under the curve of \(f(x)\) from \(x = 0\) to \(x = 5\). Since \(f(x)\) is decreasing, the average value of the function over this interval will be less than the initial value \(f(0)\) but more than the final value \(f(5)\).<br /><br />Without specific values for \(f(x)\), we can only estimate based on the options provided. If the table suggests that the values of \(f(x)\) decrease significantly from \(x = 0\) to \(x = 5\), then the integral should reflect an average value that is lower than the midpoint between the highest and lowest values of \(f(x)\).<br /><br />Given the options:<br />- A: 235<br />- B: 275<br />- C: 310<br />- D: 400<br /><br />If the function is decreasing, the integral's value should not be too high. Option D (400) seems unlikely because it would suggest a very high average value, inconsistent with a decreasing function unless the starting value was extremely high. <br /><br />Option A (235) is plausible if the function decreases steadily and significantly. Without exact values, it's reasonable to assume that the integral's value is closer to the lower end of the range provided by the options.<br /><br />Therefore, the most likely value for \(\int_{0}^{5} f(x) \, dx\) given that \(f(x)\) is decreasing is:<br /><br />A: 235