Question

jesaki inc estimates that it will sell units of product after spending $ thousand on advertising, as given by N(x)= 0.43⁴ + 47x³- 1,056x²+ 158,684. what is the point of diminishing returns? round to the nearest dollar. $

Answer 1

After spending $1,229.48 on advertising, any additional investment will lead to a smaller increase in sales compared to previous investments.

How to solve

To find the point of diminishing returns for Jesaki Inc., we need to identify the advertising expenditure (x) where the marginal increase in sales (N'(x)) starts to become negative.

Here's how we can do this:

Find the Marginal Sales Function: The marginal sales function, N'(x), represents the rate of change of sales (N) concerning advertising expenditure (x). We can find it by taking the derivative of N(x):

N'(x) = 1.72x^3 + 141x^2 - 2,112x

Identify the Critical Point: The point of diminishing returns occurs where N'(x) = 0. We can solve this equation to find the critical value of x:

`1.72x^3 + 141x^2 - 2,112x = 0\\x(1.72x^2 + 141x - 2,112) = 0`

x = 0 or x = 1,229.48 (discard x = 0 since it doesn't make sense in this context)

Round and Interpret: Therefore, Jesaki Inc. reaches the point of diminishing returns at an advertising expenditure of $1,229.48 rounded to the nearest dollar.

This means after spending $1,229.48 on advertising, any additional investment will lead to a smaller increase in sales compared to previous investments.

The Complete Question

Jesaki Inc. estimates its product sales, measured in units, based on advertising spending, represented by the function N(x) = 0.43x^4 + 47x^3 - 1,056x^2 + 158,684.

At what advertising expenditure does Jesaki Inc. reach the point of diminishing returns for its product sales? Provide the value rounded to the nearest dollar.