A company has a price demand function given by p 3 x 60...

A company has a price-demand function given by p = -3√x + 60, where x is the number of items sold at a price of $p per item. The company also has a cost function given by C(x) = 90√x + 12x + 5000 dollars when x items are made. If P(x) represents the company's profit function, in dollars, when x items are made and sold, find P'(144). Interpret your answer.

Transcript

00:01 Okay, so you have a price function given by when prices p, x items are sold.

00:07 So negative 3 square root of x plus 6.

00:15 And therefore, your revenue function, because profit is revenue minus cost, you need to know your revenue function.

00:22 It's going to be x times negative 3 square root of x plus 6.

00:28 It's just the number that you sell times what price it is.

00:32 So your revenue function, if you distribute that x through, gives you negative 3.

00:39 So this is x to the 1 half.

00:41 This is x to the 2 halves.

00:46 Maybe i'll write that as x to the 1 half.

00:52 So i can write it as a fractional exponent.

00:56 So i can write it as negative 3x to the 3 halves plus 6x.

01:05 So that's your revenue function.

01:06 They give you a cost function that's given by 90 square root of x.

01:16 Well, let me write that as x to the one -half.

01:19 I'd like to write things as fractional exponents when i can instead of roots because it makes using the power rule a little bit easier than plus 12x plus 5.

01:29 That's your cost function.

01:31 So profit, trying to make it a, maybe i'll make it a fancy p to differentiate it from price.

01:40 So profit is revenue minus cost.

01:53 And that's going to be negative 3x to the three halves plus 6x minus 90x to the 1 half, minus 12x minus 5.

02:09 So if i gather like terms where i can, i get a profit of x is equal to negative 3x to the 3 halves minus 6x minus 90 x to the 1 half minus 5 okay so next we have to find a derivative of the profit function so in this case we just have to use the power rule to get that um at well let's uh evaluate this the derivative first so to get p prime of x we just do the power rule to get negative 3 times 3 over 2 x to the subtract 1 from the exponent to get 1 half and get a minus 6 then a minus 90 times 1 half x to the negative 1 half and again we just took the exponent and minus 1.

03:14 Hopefully you know how to use the power rule.

03:17 Then minus 5 that's a constant so it's derivative is 0.

03:20 So this is our derivative for profit right here.

03:25 That's our derivative for profit.

03:27 Let's clean it up a little bit if we can...

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