4. The profit $P$ (in millions of dollars) for a T-shirt company can be modeled by $P=-x^{3}+4 x^{2}+x$ where $x$ is the number of T-shirts (in millions) of T-shirts produced. Currently the company produces 4 million $T$-shirts and makes a profit of $\$ 4$ million. What lesser number of $T$-shirts could the company produce and still make the same profit?
Answer
Since we need a lesser number of T-shirts than 4 million, the company could produce \( \boxed{1} \) million T-shirts and still make the same profit of $4 million.
Step 1 ：Given the profit function P(x) = -x^3 + 4x^2 + x, we need to find the value of x when P(x) = 4 and x < 4.
Step 2 ：Solve the equation: \( -x^3 + 4x^2 + x = 4 \)
Step 3 ：The solutions are x = -1, 1, and 4.
Step 4 ：Since we need a lesser number of T-shirts than 4 million, the company could produce \( \boxed{1} \) million T-shirts and still make the same profit of $4 million.
