Determine whether rolle's theorem applies to the function shown below on the given interval. if so, find the point(s) that are guaranteed to exist by rolle's theorem. f(x)equals=x(xminus−44)squared2; [0,44]

Since the function is continuous between x = 0 and x = 44 then Rolle's theorem applies here.

Differentiating

y' = x * 2(x - 44) + (x - 44)^2

y' = 3x^2 - 176x + 1936 = 0 (at a turning point).

solving we get x = 44 , 14.67

y" = 6x - 176 which is negative for x = 14.67 so this gives a maximum value for f(x)

This maximum is at the point (14.67, 12,619.85)

There is a minimum at ( 44,0)

These are the required points

1. Create a ratio of two products. In the numerator, start with the number of total numbers (in your case 10) and count down so that total number of items being multiplied is equal to the number of items being selected (in your case three numbers should be multiplied). In the denominator, start with the number of items being selected and count down to 1 (in your case start from number 3).

Therefore, three students from 10 can be selected in different orders.
2. The probability of rolling a number of 8 with two dice is the number of times that number 8 may occur divided by 36 (6*6 - number of all possibilities  for two dice). You can obtain a sum of 8 in these cases: 6+2, 5+3, 4+4, 3+5, 2+6, then the probability is .