If a price-demand equation is solved for p, then price is expressed as p = g(x) and x becomes the independent variable. In this case, it can be shown that the elasticity of demand is given by E(x) = - [g(x) / xg'(x)]. Use the given price-demand equation to find the values of x for which demand is elastic and for which demand is inelastic.

p = g(x) = 12960 - 0.1x^2

I am really stuck on this question and don't know how to start, could someone please show me?

you have to remember that inelastic means that E(x) > 0

(I think)

E(x) = -[(12960-.1x^2)/(-.2x^2)]
= 64800/x^2 - .5

E(x) > 0 means x < 360

That sound familiar? Or am I way off base here?

To find the values of x for which demand is elastic and for which demand is inelastic, we need to determine the elasticity of demand function, E(x), using the given price-demand equation, p = 12960 - 0.1x^2.

The elasticity of demand, E(x), can be calculated using the formula:
E(x) = - [g(x) / x * g'(x)]

Let's start by finding g'(x), the derivative of g(x) with respect to x:
g(x) = 12960 - 0.1x^2
g'(x) = -0.2x

Substituting the values of g(x) and g'(x) into the elasticity of demand formula, we have:
E(x) = - [(12960 - 0.1x^2) / (x * -0.2x)]

Simplifying, we get:
E(x) = (12960 - 0.1x^2) / (0.2x^2)

Now, to determine when demand is elastic and when it is inelastic, we need to consider the magnitude of the elasticity of demand. If E(x) > 1, demand is elastic. If E(x) < 1, demand is inelastic.

Setting E(x) = 1 and solving for x will give us the dividing point between elastic and inelastic demand.

1 = (12960 - 0.1x^2) / (0.2x^2)

Cross-multiply:
0.2x^2 = 12960 - 0.1x^2

Combine like terms:
0.3x^2 = 12960

Divide by 0.3:
x^2 = 43200

Take the square root of both sides:
x = ± 207.887

Since price cannot be negative, we consider only the positive value:
x = 207.887

Therefore, demand is elastic for values of x greater than 207.887 and inelastic for values of x less than 207.887.