To determine whether the revenue earned from sales of shoes and accessories is increasing or decreasing, we need to analyze the first derivative of the revenue function [tex]\( s(x) \)[/tex].
Given the revenue function:
[tex]\[ s(x) = 0.6x^3 - 10.2x^2 + 32.4x + 127.2 \][/tex]
First, we find the first derivative of the function [tex]\( s(x) \)[/tex]:
[tex]\[ s'(x) = \frac{d}{dx}(0.6x^3 - 10.2x^2 + 32.4x + 127.2) \][/tex]
[tex]\[ s'(x) = 1.8x^2 - 20.4x + 32.4 \][/tex]
Next, we evaluate the first derivative at [tex]\( x = 4.5 \)[/tex] to determine if the revenue is increasing or decreasing at this value of [tex]\( x \)[/tex]:
[tex]\[ s'(4.5) = 1.8(4.5)^2 - 20.4(4.5) + 32.4 \][/tex]
[tex]\[ s'(4.5) = 1.8 \cdot 20.25 - 20.4 \cdot 4.5 + 32.4 \][/tex]
[tex]\[ s'(4.5) = 36.45 - 91.8 + 32.4 \][/tex]
[tex]\[ s'(4.5) = -22.95 \][/tex]
Since [tex]\( s'(4.5) = -22.95 \)[/tex], this result indicates that the slope of the revenue function [tex]\( s(x) \)[/tex] is negative at [tex]\( x = 4.5 \)[/tex].
Therefore, between the 3rd and 6th months, the revenue earned from sales of shoes and accessories is decreasing. Accordingly, we fill in the given statement as follows:
Between the 3rd and 6th months, the revenue earned from sales of shoes and accessories is [tex]\(\checkmark\)[/tex] decreasing at [tex]\( -22.95 \)[/tex].
