Entry 9-Assignment 3

The assignment is due 1st December. The first four questions are not that hard and they are similar to the questions that Danny did in the class.

However, the last two questions are very challenging and hard. First of all, I have to decide whether to prove or disprove the statement. I pick the side of disapproval at first. I try to find two functions such that one does grow faster than and not grow slower than the other function. I try to find an equation that in some interval, one function is growing faster than the other; then in other interval, the one is growing slower than the other. Following this idea, I successfully find that trig functions fully satisfy my requirements. Now the questions becomes that how can I disprove the statement using the two trig functions, sinx and cosx.

I spend 1 hour to prove the negation of given statement. I realize the sin and cos are  not good example. Therefore I give up trig function and try to come up with another examples. Trig functions are continuous functions, and one-piece continuous functions are hard to approach in this problem, so I try to find two-pieces functions that satisfy the condition.

Eventually, I find one!!!! Even though this question takes my 2 hours, I feel it is worthwhile！！！