Acompany analyzed its stock price for the period 2002 to 2010 and modeled the price as a polynomial given by p(t) = t3 – 12t2 + 32t + 50, where p is the stock price (in dollars) and t is the number of years passed since january 1, 2002. in which period will the stock price be lower than $50?

Given that the stock price of a given company over a period of 8 years can be modeled by P(t)=t^3-12t^2+32t+50. Where p is the price and t is the number of years passed.
To get the period whereby the stock price will be less than $50 we proceed as follows;
First we find the derivative of the function;
p'(t)=3t^2-24t+32=0
thus solving for t we get;
t=4+\-4/sqrt3
or
t=6.31 or 1.69
Evaluating the values of p at this point we get:
p(1.69)=74.63
p(6.31)=25.36
Evaluating the point before  and after t=1.69 say t=0 and t=3 we get:
p(0)=50
p(2)=65
Evaluating the point immediately before and after t=6.31 say t=6 and t=7
p(6)=26
p(7)=29
from the above we see that the lowest point was at point t=6.31, thus the time period when t was below $50 was at the interval t=0 and t=6.31