Question

The profit of a company can be modelled by the polynomial function p(t)= -t^3 +12t^2 -21t +10 , where p is the profit, in thousands of dollars and t is the time, in years. When will the company make their maximum profit of $108000?

Answer 1

find the critical points of the function p(t):

$$\frac {d p(t)} {dt}=0=-3t^2+24t-21$$ $$t_{1,2}=\frac {-24\pm \sqrt{24^2-4(-3)(-21)}} {2(-3)}$$ $$t_1=1, \quad t_2=7$$

second derivative test:

$$\frac {d^2 p(t)} {dt^2}=-6t+24$$ $$-6t_1+24=18>0 \quad \to \quad t_1=1-minimum$$ $$-6t_2+24=-18<0 \quad \to \quad t_2=7-maximum$$

function value:

$$p(t_2)=-(7)^3+12(7)^2-21(7)+10=108$$

Answer: the company will make a maximum profit of $108,000 in 7 years