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

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