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Question 46 of 58

Q.If x + y = 60 find the maximum value of xy³.

West Bengal WbchseWest Bengal HS (WBCHSE) Board 2023Subjective· 5mImportance★★★★★
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Reduce to a one-variable function using y=60−xy=60-x, then use the first-derivative test to locate and confirm the maximum.

Step 1 — reduce to one variable. Since x+y=60x+y=60, write y=60−xy=60-x and f(x)=xy3=x(60−x)3f(x)=xy^3=x(60-x)^3.

Step 2 — differentiate.

f′(x)=(60−x)3+x⋅3(60−x)2(−1)=(60−x)2[(60−x)−3x]=(60−x)2(60−4x).f'(x)=(60-x)^3+x\cdot3(60-x)^2(-1) = (60-x)^2\big[(60-x)-3x\big] = (60-x)^2(60-4x).

Step 3 — find critical points. f′(x)=0f'(x)=0 when (60−x)2=0(60-x)^2=0 (i.e. x=60x=60, giving y=0y=0, a boundary/minimum with f=0f=0) or when 60−4x=060-4x=0, i.e. x=15x=15.

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