Question 721098
The total population is {{{f(t)=100e^t+1000 e^(-t)}}}
The derivative of that function would be the population growth rate
{{{g(t)=100e^t-1000 e^(-t)}}}
When that growth is zero the population will have reached its minimum:
{{{100e^t-1000 e^(-t)=0}}}
If we define {{{x=e^t}}}  then {{{e^(-t)=1/x}}} and we can solve
{{{100x-1000/x=0}}} as a first step
{{{100x-1000/x=0}}} --> {{{(100x^2-1000)/x=0}}} --> {{{100x^2-1000=0}}} --> {{{x^2-10=0}}} --> {{{x^2=10}}} --> {{{x=sqrt(10)}}}
In terms of {{{t}}}
{{{e^t=sqrt(10)}}} --> {{{t=ln(sqrt(10))}}} --> {{{t=ln(10^0.5)}}} --> {{{t=0.5ln(10)}}} or {{{t=ln(10)/2}}}= approx {{{1.151}}}
So the total deer population will reach its minimum after about 1.151 years.