x & 0 & 3 & 6 & 9 & 12 & 15 f(x) & 0 & 2 & 6 & 12 & 20 & 30 The table above gives selected values for a continuous function f . If f is increasing over the closed interval [0,15] , which of the following could the value of int_(0)^15 f(x) d x ? A. 70 B 120 C. 160 D. 210

Given that \( f \) is an increasing function over the interval \([0, 15]\), we can estimate the integral \(\int_{0}^{15} f(x) \, dx\) by considering the values of \( f(x) \) at the endpoints and midpoints of subintervals.<br /><br />Since \( f \) is increasing, the value of the integral will be greater than the area under the curve formed by the minimum value of \( f(x) \) (which occurs at \( x = 0 \)) and less than the area under the curve formed by the maximum value of \( f(x) \) (which occurs at \( x = 15 \)).<br /><br />If we assume \( f(x) \) starts at a lower value and increases to a higher value, the integral should reflect this increase. Given the options:<br /><br />A. 70<br />B. 120<br />C. 160<br />D. 210<br /><br />The most reasonable estimate for the integral \(\int_{0}^{15} f(x) \, dx\) given that \( f \) is increasing would likely be around the middle range of the provided options. <br /><br />Therefore, the correct answer is:<br />C. 160