Indicate which of the following mathematical theorem can best be used to justify the given statement.

1) Chain rule

2) Extreme value theorem

3) Fundamental Theorem of calculus

4) Intermediate value theorem

5) Mean value theorem

a) A Tibetan monk leaves the monastery at 6 am and takes his usual path to the top of the mountain, arriving at 6 pm. The following morning, he starts at 6 am at the top of the mountain, and takes the same path back, arriving at the monastery at 6 pm. There is a point on the path that the monk will cross at exactly the same time of day on both days.

The theorem is ___________

b) Let v(t) be the velocity of a particle at time t. If s(t) is the distance traveled from time t=0 then,

s(t)= integral (0 to t )v(x)dx

The theorem is ___________

The size of bacteria population depends on the availability of the nutrients as food. The rate at which the bacteria population is changing with time is the product of the rate at which availability of nutrients changes with time and the rate at which the bacteria population changes in reaction to the change in the nutrient availability

The theorem is ___________