We let the derivative equals to zero and the guards great values presenting the Humira. And if they want the maurico approximation, we can use a calculator for per seat, the local maximum and minimum. But just evaluate the function value of the derivative function at this three values and which gives us these three values. The derivative of Kansai X equals minor Sign X and as a derivative of party over 26 equals pi over 26 for part B. To take the derivative of this profit function, we need to use the chain rule. We are asked to find the rate of the change of the prophet, which is just the first order derivative of the profit function. So this is the rate off change off profit Art Execute 20 and DX. So there's 400 minus 80 which comes orders 3 20. The rate of change of revenue, which is here on the rate of change of cost, so profit will just be the subtraction of both.
So the rate of change of profit as a DP or D D as will be D R or D D on D c or D T so, uh, attacks at these points, which were given at X equals 20 and D X or D t as 20. We know that profit as revenue minus cost. So this is the rate of change off cost, and the last one is the rate of change of profit. D has already given us 20 so pretty times four is 80. And so if we differentiate both sides that respect o t uh, we have four disasters.
Let's talk about part B, which is the cost. So let's put access 30 if x 30 than 50 minus 30 is 20 and this is already 20. We eloped with 50 minus 500.5 times to it once we just have X. So we'll say that D R or D t will be 50 on this will be d X over d t over here on 0.5 Disasters The transition of excess Where will be two x t x over d. So let's differentiate are both sides with respect o t. We are given revenue and cost functions, and we need to find the rate of change off revenue costs and profit functions at X equal to 10 and the X over d t r 20.
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