(1 point) For the cost function $$ C(x)=1350+470 x+0.4 x^{2} $$ and the demand function $$ p(x)=1410 $$, find the production lovel that will maximaze profit. Production Lovel $$ = $$

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We first compute the revenue function $$ R(x) $$ from the demand function:
$$
R(x)=p(x)x=1410x.
$$

Next, the profit function $$ \pi(x) $$ is given by revenue minus cost:
$$
\pi(x)=R(x)-C(x)=1410x-(1350+470x+0.4x^2).
$$

Simplify $$\pi(x)$$:
$$
\pi(x)=1410x-470x-0.4x^2-1350,
$$
$$
\pi(x)=940x-0.4x^2-1350.
$$

This profit function is a quadratic in $$ x $$ of the form:
$$
\pi(x)=-0.4x^2+940x-1350.
$$

Since the coefficient of $$ x^2 $$ is negative, the quadratic opens downward and the profit is maximized at the vertex.

The $$ x $$-coordinate of the vertex is found using:
$$
x=-\frac{b}{2a}.
$$

Here, $$ a=-0.4 $$ and $$ b=940 $$. Thus,
$$
x=-\frac{940}{2(-0.4)}=-\frac{940}{-0.8}=\frac{940}{0.8}=1175.
$$

The production level that maximizes profit is $$ \boxed{1175} $$.
The production level that maximizes profit is 1175.

(1 point)
For the cost function and the demand function , find the production lovel that will maximaze profit.
Production Lovel

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(1 point) For the cost function and the demand function , find the production lovel that will maximaze profit. Production Lovel