00:01
In this question, we are asked to maximize the profit function.
00:04
And the first step is to solve the equation p ' of x equals 0, where p ' is the derivative of the function p.
00:14
So, the derivative of, to get the derivative of p, we need to differentiate ln first.
00:19
The derivative of ln is 1 over the expression inside parentheses, and then we need to multiply that by the derivative of the expression in the numerator, sorry, by the derivative of the expression inside the logarithm.
00:37
That's going to be negative 3x squared plus 6x plus 105.
00:47
Then p ' is 0 if negative 3x squared plus 6x plus 105 is 0.
01:00
Let's divide everything by negative 3.
01:02
We'll get x squared minus 2x minus 35 equals 0.
01:09
That's a quadratic equation.
01:11
The discriminant is 4 plus 4 times 35.
01:18
That's going to be 144.
01:20
Then the roots are x equals to 2 plus minus the square root of 144 divided by 2...