4. (5pts) The profit function is defined to be the revenue minus the cost at the production level $x$. That is, the profit function $P(x)=R(x)=C(x)$. Use your previous work to find $P(x)$, then sketch the graph of this function.
Hint: use the domain found in Problem 1. $R(x)=6400x-0.002x^{\wedge }2$ and $C(x)=85x+50000$
Domain $=[0,3200000]$

Step 1 ：First, we need to find the profit function $P(x)$, which is the revenue function $R(x)$ minus the cost function $C(x)$.

Step 2 ：Given that $R(x)=6400x-0.002x^2$ and $C(x)=85x+50000$, we can substitute these into the profit function.

Step 3 ：So, $P(x)=R(x)-C(x)=(6400x-0.002x^2)-(85x+50000)$.

Step 4 ：Simplify the equation to get $P(x)=6315x-0.002x^2-50000$.

Step 5 ：Next, we need to sketch the graph of this function. We can do this by finding the roots of the equation, the vertex, and the y-intercept.

Step 6 ：The roots of the equation are the values of $x$ when $P(x)=0$. So, we set $6315x-0.002x^2-50000=0$ and solve for $x$.

Step 7 ：However, this equation is a quadratic equation and it's a bit complex to solve. So, we can use the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, where $a=-0.002$, $b=6315$, and $c=-50000$.

Step 8 ：Substitute these values into the quadratic formula to get $x=\frac{-6315\pm \sqrt{{6315}^2-4\ast (-0.002)\ast (-50000)}}{2\ast (-0.002)}$.

Step 9 ：Solving this equation gives us two roots, but we only consider the positive root because the quantity produced cannot be negative.

Step 10 ：The vertex of the parabola is given by the formula $x=-\frac{b}{2a}$, where $a=-0.002$ and $b=6315$. Substituting these values gives us $x=-\frac{6315}{2\ast (-0.002)}$.

Step 11 ：The y-intercept is the value of $P(x)$ when $x=0$. So, we substitute $x=0$ into the equation to get $P(0)=-50000$.

Step 12 ：With these points, we can sketch the graph of the function $P(x)=6315x-0.002x^2-50000$.

Step 13 ：Finally, we check whether our results meet the requirements of the problem. The graph should be a downward-opening parabola with the vertex at the point calculated above, and it should intersect the x-axis at the roots calculated above. The y-intercept should be at $P(0)=-50000$. The graph should also be within the domain $[0,3200000]$.

Step 14 ：Thus, the profit function $P(x)$ and its graph have been found.