If the profit function is $P ( x ) = \frac { 5 x ^ { 2 } } { 3 } + \frac { 2 } { 3 x } + 2 x - 10$ for producing a product, find the marginal profit at a production level of 10 units.
A) $\frac { 5299 } { 150 }$
B) $\frac { 2651 } { 150 }$
C) 60
D) 423
E) $\frac { 252 } { 75 }$

Question

Quizplus · Accepted Answer

To find the marginal profit at a production level of 10 units, we need to differentiate the profit function $P(x)$ with respect to $x$ and then substitute $x = 10$. The profit function is $P(x) = \frac{5x^2}{3} + \frac{2}{3x} + 2x - 10$. Differentiating $P(x)$ with respect to $x$ gives us the marginal profit function $P'(x) = \frac{10x}{3} - \frac{2}{3x^2} + 2$. Substituting $x = 10$ into $P'(x)$ gives $P'(10) = \frac{10(10)}{3} - \frac{2}{3(10)^2} + 2 = \frac{100}{3} - \frac{2}{300} + 2 = \frac{3000}{300} - \frac{2}{300} + \frac{600}{300} = \frac{3598}{300} = \frac{5299}{150}$.

If the Profit Function Is P(x)=5x23+23x+2x−10P ( x ) = \frac { 5 x ^ { 2 } } { 3 } + \frac { 2 } { 3 x } + 2 x - 10P(x)=35x2​+3x2​+2x−10

If the Profit Function Is $P ( x ) = \frac { 5 x ^ { 2 } } { 3 } + \frac { 2 } { 3 x } + 2 x - 10$