Question

A firm estimates that it can sell Q units of its product with an advertising expenditure of x thousand dollars where = ( ) =− ଶ +600 +25
i) Over what level of advertising expenditure is the number of units of product sold increasing?
ii) Over what level of advertising expenditure is the number of units of product sold decreasing?​

Answer 1

SOLUTION

GIVEN

A firm estimates that it can sell Q units of its product with an advertising expenditure of x thousand dollars

$\sf{Q(x) = - {x}^{2} + 600x + 25}$

TO DETERMINE

i) Over what level of advertising expenditure is the number of units of product sold increasing?

ii) Over what level of advertising expenditure is the number of units of product sold decreasing?

EVALUATION

Here it is given that the firm estimates that it can sell Q units of its product with an advertising expenditure of x thousand dollars

$\sf{Q(x) = - {x}^{2} + 600x + 25}$

Differentiating both sides with respect to x we get

$\sf{Q'(x) = - 2x + 600}$

(i) Since Q(x) is increasing

$\sf{Q'(x) > 0}$

$\sf{ \implies \: - 2x + 600 > 0}$

$\sf{ \implies \: - 2x + 600 > 0}$

$\sf{ \implies \: - 2x > - 600 }$

$\sf{ \implies \: x < 300 }$

The required amount is less than 300 dollars

(ii) Since Q(x) is decreasing

$\sf{Q'(x) > 0}$

$\sf{ \implies \: - 2x + 600 < 0}$

$\sf{ \implies \: - 2x < - 600}$

$\sf{ \implies \: x > 300}$

The required amount is greater than 300 dollars

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