Question 1

According to the law of diminishing returns, what happens to the rate of profit as investment in a single input increases while other inputs remain constant?

Accepted Answer

The rate of profit increases up to a point, then begins to decrease.

Question 2

Given the sales function $$N(x) = 3x^2$ - 0.25x^4$ + 200$$, where $$x is $$the advertising expenditure in thousands of dollars, what is the derivative $$N'(x)$$ representing?

Accepted Answer

$$N'(x)$$ represents the rate of change of sales with respect to advertising expenditure.

Question 3

At what value of $$x$$ does the point of diminishing returns occur for the function $$N(x) = 3x^2$ - 0.25x^4$ + 200$$?

Accepted Answer

$$x = 6$$

Question 4

What is the economic interpretation of the inflection point on the sales curve $$N(x)$$ with respect to advertising expenditure?

Accepted Answer

It is the point where the rate of change of sales switches from increasing to decreasing.

Question 5

If a company spends $$6,000 on advertising, what is the expected rate of increase in sales according to the model N(x$$)$?

Accepted Answer

108 TVs per thousand dollars spent

Question 6

Which of the following best describes the behavior of $$N'(x)$$ before and after the point of diminishing returns?

Accepted Answer

$$N'(x) is $$increasing before the point and decreasing after the point.

Question 7

Suppose a farmer increases fertilizer use on a fixed plot of land. According to the law of diminishing returns, what will eventually happen to the additional crop yield per unit of fertilizer?

Accepted Answer

It will decrease after a certain point.

Question 8

Given $$N'(x) = 9x^2$ - x^3$, what is N$$''(x)$$?

Accepted Answer

$$N''(x) = 18x - 3x^2$

Question 9

If $$N''(x) = 0 at$$ $$x = 6$$, what does this indicate about the function $$N(x) at$$ $$x = 6$$?

Accepted Answer

There is an inflection point at $$x = 6.$$

Question 10

A company increases its advertising budget from $$5,000 to 7$$,000. According to the law of diminishing returns, what is the most likely outcome?

Accepted Answer

Sales will increase, but at a lower rate than before.

Question 11

Which calculus concept is most directly used to identify the point of diminishing returns on a production or sales curve?

Accepted Answer

Finding where the second derivative is zero (inflection point).

Question 12

If $$N(x)$$ has an inflection point at $$x = 6$$, what does this mean for the concavity of $$N(x)$$?

Accepted Answer

$$N(x)$$ changes from concave up to concave down at $$x = 6.$$

Question 13

A business wants to maximize the rate of change of its sales with respect to advertising. What value should it choose for $$x in $$the model $$N(x) = 3x^2$ - 0.25x^4$ + 200$$?

Accepted Answer

$$x = 6$$

Question 14

Which of the following best describes the law of diminishing returns in the context of advertising?

Accepted Answer

After a certain point, additional advertising yields smaller increases in sales.

Question 15

If $$N(x)$$ represents the number of TVs sold per month and $$x is $$the advertising expenditure in thousands of dollars, what does $$N(6)$$ represent?

Accepted Answer

The number of TVs sold when $6,000 is spent on advertising.

The Point of Diminishing Returns in Business Calculus

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