I have to determine the price that needs to be charged to obtain the largest revenue.
q=-p^2+33p+9 so
R=p(-p^2+33p+9) or =-p^3+33p^2+9p
I then need to take the derivative
R'=-3p^2+66p+9
Now I somehow need to solve for p to determine my price to obtain my largest revenue. How do I do this.
To find a maximum, set your marginal revenue (R') equal to zero. Now then, R' is a quadradic equation. So, use the quadradic formula.
-b +- sqrt(b^2+4ac) / 2a
my bad. it's -4ac in the quadradic, not +4ac
To determine the price that will result in the largest revenue, you need to find the value of p (price) where the derivative R' equals zero or where the derivative changes from positive to negative (reaching a maximum point). Here's how you can solve for p:
1. Start with the derivative of the revenue function: R' = -3p^2 + 66p + 9.
2. Set R' equal to zero: -3p^2 + 66p + 9 = 0.
3. Now, you can solve this quadratic equation for p. There are a few different methods you can use to solve this, such as factoring, completing the square, or using the quadratic formula.
- Factoring (if possible):
You would try to factor the quadratic equation into two binomial factors. However, in this case, factoring might not be straightforward because the coefficients of p^2 and the constant term are not simple numbers.
- Completing the square:
Complete the square by manipulating the equation to look like (p - h)^2 = k. However, completing the square might also not be the most efficient method in this case due to the coefficients.
- Quadratic formula:
The quadratic formula gives the solutions to any quadratic equation of the form ax^2 + bx + c = 0. The formula is x = (-b ± √(b^2 - 4ac))/(2a).
Applying this formula to the quadratic equation -3p^2 + 66p + 9 = 0, you can find the solutions for p.
4. Once you find the values of p where R' = 0 (or the critical points), you can then determine which value corresponds to the maximum revenue by evaluating the second derivative test or by comparing the function values at the critical points.
- Second derivative test:
Take the second derivative of the revenue function, R'': R'' = -6p + 66. Evaluate R'' at each of the critical points found in step 3. If R'' is positive at a critical point, then it corresponds to a minimum, whereas if R'' is negative, it corresponds to a maximum. In this case, since you want the largest revenue, you are interested in the maximum point.
- Evaluating function values:
Plug the critical points into the original revenue function q=-p^2+33p+9 (or R) and see which value results in the maximum revenue.
Once you identify the price (p) that corresponds to the maximum revenue, you have determined the price that needs to be charged to obtain the largest revenue.