Hi, I had this question, but I'm unable to solve it without using derivatives, though I have the answer (which is $15.7$ approx). I would appreciate it if you could help me out how to get the answer without using derivatives. Thanks a lot.
Question: The profit of a company can be modelled by the polynomial function $p(t)= -4t^3+10t^2+8t-6$, 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 $18 000 $ $?
The answer, believe it or not, is actually $2$.
What you (probably) did was get the equation $-4t^3+10t^2+8t-6 = 18000$. However, this is not correct, because of the caveat in the problem statement. It says "$p$ is the profit in thousands of dollars."
So we express $18000$ as $18$, since we are in thousands of dollars. The equation is therefore $$-4t^3+10t^2+8t-6=18.$$
This simplifies to $4t^3-10t^2-8t+24.$ Note that we can divide by $2$ to get $2t^3-5t^2-4t+12$.
To solve this, use the Rational Root Theorem. Checking $1, -1, 2, -2, \dots$ note that $t = 2$ is a solution by RRT.
Dividing $2t^3-5t^2-4t+12$ by $t-2$ gets $2t^2-t-6$. The two solutions to this are $t = 2$ and $t = -\frac{3}{2}$.
So the company will make their maximum profit in $2$ years.