Science:Math Exam Resources/Courses/MATH101/April 2005/Question 08 (b)
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Question 08 (b)
An unknown continuous function satisfies , , and
for Also, is nondecreasing on this interval, i.e. it satisfies for all real numbers and with Let be the value of definite integral
(b) Find the smallest and largest posible values for . Give explicit functions , satisfying the conditions above, that yield these smallest and largest values.
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The minimum is given by and the maximum is given by
To show that is the minimum, we note that our is a nondecreasing function, and we also require that . This last condition means that the absolute smallest can be is and it turns out that this function also has the first two desired properties so this function works and is minimal.
For the second function, we note that our is a nondecreasing function; we also have and lastly that . Again to make the function maximal and nondecreasing, we really would like however this breaks the upper bound condition. So we pick the function until we reach the value of 8 and then pick the function until the end where we also need to hope that on this interval (so that we are not breaking the upper bound condition. Let . Notice that
This means that the function is non-decreasing until . Furthermore, notice that
when and so this occurs when and . Notice that since this is a parabola, it has only one critical point (which we computed to be at and so indeed between , we have that
Thus, the function chosen above has the desired properties we require and thus gives the maximum value for I. Computing these values
completing the question.