When an electronics store prices a certain brand of stereos at p hundred dollars per set, it is found that q sets will be sold each month, where q2 + 2 p2 = 41.

a. Find the elasticity of demand for the stereos using implicit differentiation.
b. For a unit price of p = 4 ($400), is the demand elastic, inelastic, or of unit elasticity?
c. Find the number of stereo sets that will maximize revenue (prove your answer).

mr giloni will find you!!!

a. To find the elasticity of demand using implicit differentiation, we need to find the expression for the quantity demanded, q, in terms of the price, p. The given equation, q^2 + 2p^2 = 41, represents the relationship between price and quantity demanded.

To differentiate both sides of the equation with respect to p, we get:

2q(q') + 4p(p') = 0

Rearranging the equation to solve for q', the derivative of q with respect to p, we have:

q' = -2p(p') / (2q)

Now, to find the elasticity of demand, we need to determine the ratio of the relative change in quantity demanded to the relative change in price. This can be expressed as:

E = (dq / q) / (dp / p)

Using the chain rule, we can rewrite dq/dp as (dq/dp) * (dp/dp), which simplifies to dq/dp.

Thus, the elasticity of demand can be calculated as:

E = (dq / q) / (dp / p) = q' * (p / q)

Substituting the expression for q' that we obtained earlier:

E = (-2p(p') / (2q)) * (p / q) = -p(p') / q

b. Given that the unit price is p = 4 ($400), we need to determine if the demand is elastic, inelastic, or of unit elasticity. To find this, we need to evaluate the elasticity of demand at this price.

Using the expression for the elasticity of demand derived above, we can substitute p = 4 and q from the given equation q^2 + 2p^2 = 41:

E = -p(p') / q = -4(p') / q

To determine the sign of the elasticity, we need to know whether p' (the derivative of p with respect to q) is positive or negative. However, we don't have information regarding the relationship between price and quantity demanded, so we cannot determine the sign of p'. Therefore, we cannot conclusively determine whether the demand is elastic, inelastic, or of unit elasticity.

c. To find the number of stereo sets that will maximize revenue, we need to consider the revenue function. Revenue is calculated by multiplying quantity demanded (q) with the price (p):

R = p * q

Since we're given the equation q^2 + 2p^2 = 41, we can solve for q in terms of p:

q^2 = 41 - 2p^2
q = sqrt(41 - 2p^2)

Now, substitute this expression for q into the revenue function:

R = p * sqrt(41 - 2p^2)

To maximize revenue, we differentiate R with respect to p, set the derivative equal to 0, and solve for p:

R' = sqrt(41 - 2p^2) - 2p^2 / sqrt(41 - 2p^2) = 0

Simplifying the equation:

1 - 2p^2 = 0

2p^2 = 1

p^2 = 1/2

p = ± sqrt(1/2)

Since we are dealing with a positive price value, p must be p = sqrt(1/2).

Finally, substitute the value of p back into the equation for q:

q = sqrt(41 - 2p^2) = sqrt(41 - 2(1/2)) = sqrt(40) = 2sqrt(10)

Therefore, the number of stereo sets that will maximize revenue is 2sqrt(10).