Consider the printed area of horizontal length "x" and vertical length "y" as shown in the figure below:
Area of printed region as shown above in the figure is,
Here, the objective is to minimize the total area of the poster"A=(x+4)(y+8)"
Substitute "\\frac{32}{x}" for "y" into "A=(x+4)(y+8)" to express the total area in terms of single variable "x" as,
Differentiate with respect to "x" and equate to zero in order to find the critical value as,
"A'(x)=0"
"(x+4)\\frac{d}{dx}(\\frac{32}{x}+8)+(\\frac{32}{x}+8)\\frac{d}{dx}(x+4)=0"
"(x+4)(-\\frac{32}{x^2})+(\\frac{32}{x}+8)(1)=0"
"-\\frac{32}{x}-\\frac{128}{x^2}+\\frac{32}{x}+8=0"
"-\\frac{128}{x^2}+8=0"
"x^2=16"
"x=4"
Next plug "x=4" into "y=\\frac{32}{x}" to obtain "y=\\frac{32}{4}=8"
Here, the first derivative change sign from negative to positive at "x=4" , therefore, there exists a minima at "x=4" .
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