Suppose at birth, every human being is given a fixed number of heartbeats, and

once these heartbeats are used up$,$ life ends. How should one optimally use these

heartbeats to have a life as long as possible? It is tempting to suggest that one is

better off staying in bed and resting, to maintain a low heartbeat. However, the

science of good health does point to the fact that a trained heart beats more slowly

when the person is at rest. On this note, consistent exercising is not a bad idea.

Suppose that the untrained heart beats $80$ times per minute when a person is at rest,

and during exercise it beats $120$ times per minute. If a person exercises the fraction

$x$ of its time, the average heartbeats per minute is as follows:

$f(x)=120x+(1-x)g(x)$

To determine an appropriate function $g(x),$ we can impose a requirement that $g(x)$

should be close to $80$ for small $x,$ which means that hardly any exercise is done.

When $x$ approaches $1,$ we assume $g(x)$ to be close to $50,$ meaning the person is well$-$

conditioned. Let's assume a simple model for $g(x)$ to be:

$g(x)=50+30e^{-100x}$

tion No$.1$ continues on page $2.$

Produce a table according to the format shown in Table $1$ in your answer script.

$($i$)$ Fill in values for $x,g(x)$ and $f(x)$ in the table. You are free to select $20$ suitable

values of $x$ to work out an approximate optimum value for exercising. State

the optimum time for exercising in minutes per day.

$($ii$)$ Write a suitable fitness function $f^{**}(x)$ to scale the fitness values so that they

are normalized from $0$ to $1.$ Fill in the values for the column labelled $f^{**}(x)$

accordingly.

$($iii$)$ From an optimization perspective, justify in your own words the level of

difficulty in solving for the optimum value of $x.$

$(13$ Marks$)$

Table $1$