Profit from marginal profit. A firm has the marginalprofit...

Profit from marginal profit. A firm has the marginalprofit function $$ \frac{d P}{d x}=\frac{9000-3000 x}{\left(x^{2}-6 x+10\right)^{2}} $$ where $P(x)$ is the profit earned at $x$ dollars per unit. Find the total-profit function given that $P=\$ 1500$ at $x=\$ 3$

Transcript

00:01 Okay, so in this question, the marginal profit function is given and we need to find the total profit function, right? so to find the total profit function, we need to take integration of marginal profit function.

00:13 So here, p is equal to integration of the given function.

00:29 Okay, now, for the numerator, we can factor out minus 3 ,000.

00:35 So we can write here minus 3 ,000, multiply by x minus 3 ,000.

00:41 And here denominator is as it is right now here minus 3 ,000 is constant so we can write it outside.

01:06 Okay now we need to substitute u is equal to x squared minus 6x plus 10 right now take the derivative on both sides so derivative of u that is to u derivative of x square is 2x derivative of minus 6x that is minus 6 derivative of 10 will be 0 so no need to write 0 multiply by dx right okay now here we can factor out two right so here we can write two multiply by x minus 3 d x now solve this equation for d x so d x is equal to d u upon 2 multiply by x minus 3 right now substitute the value of d x and u in this equation okay okay now here x minus 3 and x minus 3 will be cancelled out right and here 1 by 2 is constant so we can write it outside minus 3 ,000 divided by 2 that is equal to minus 1500 and here we have a 1 upon u square so we can write u 2 minus 2 integration of u s 2 minus 2 that is u s 2 minus 2 and we need to add constant c right and here negative sign will be cancel out and here exponent of u that is 1 so we can write in denominator right now substitute the value of you right so here you that is equal to x squared minus 6x plus 10 so we can write here x squared minus 6x plus 10 and here c that is as it is now we need to find the value of c right and for that it is given that the profit is $500 when the number of quantity is 3 right.

03:37 Now substitute these two values in this equation...

Question

Profit from marginal profit. A firm has the marginalprofit function $$ \frac{d P}{d x}=\frac{9000-3000 x}{\left(x^{2}-6 x+10\right)^{2}} $$ where $P(x)$ is the profit earned at $x$ dollars per unit. Find the total-profit function given that $P=\$ 1500$ at $x=\$ 3$

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