(a)Intermediate Value Theorem is that when a function is continuous, we can pick a number c in a interval (a,b) and there exist f(c) in a interval (f(a), f(b))  . To make someone with no knowledge of the IVT, we can give an example like taking a plane from Australia to China. As is known to all, the earth is a sphere and there is no way for us to take a plane from Australia to China without crossing the equator. No matter how pilots arrange the air line, there exist a point that he has to cross the equator and that is as the same meaning of IVT, which we can definitely find a c in a interval (a,b) and make f(c) in a interval (f(a), f(b)) if this function is continuous.

Therefore, we can know what this question want to express. Since f(0) is smaller than 0 and f(pi) is greater than 0, there exist a point a in (0, pi) such that f(a)=0.

(b)My argument is relied explicitly on the concept of continuity.

1.A parabola of a ball is a function. As is known to all, the ball will rise till its highest point and then fall down. Its kinetic trajectory is a curve and both the starting point and the ending point are equal to zero so there is no horizontal asymptote of this function.

2.Fibonacci sequence is related to golden ratio, which is used for constructing  buildings, such as calculate Proportion of a house’s area to make it more reasonable for human to live.

According to its property, the latter term equals the sum of the former two terms. Since this sequence will increase constantly, it diverges.

3.Geometric series can be used to calculate the value of an annuity.Since interest is counted monthly, using geometric series help people calculate the final value of an annuity more conveniently. People gain money every month so the rate must greater than 100%, which means the ratio is bigger than 1. Therefore, this series will diverge.