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Minima/ maxima (statement questions) - Number questions
Minima/ maxima (statement questions) - Number questions
Last updated at April 19, 2021 by Teachoo
Example 34 Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. Let first number be π Since Sum of two positive numbers is 15 π₯+ 2nd number = 15 2nd number = 15 β π Let S(π₯) be the sum of the squares of the numbers S(π₯)= (1st number)2 + (2nd number) 2 S(π)=π^π+(ππβπ)^π We need to minimize S(π) Finding Sβ(π) Sβ(π₯)=π(π₯^2+ (15 β π₯)^2 )/ππ₯ =π(π₯^2 )/ππ₯+(π(15 β π₯)^2)/ππ₯ = 2π₯+ 2(15βπ₯)(β1) = 2π₯β 2(15βπ₯) = 2π₯β30+2π₯ = 4πβππ Putting Sβ(π)=π 4π₯β30=0 4π₯=30 π₯=30/4 π=ππ/π Finding Sββ(π) Sββ(π₯)=π(4π₯ β 30)/ππ₯ = 4 Since Sββ(π)>π at π₯=15/2 β΄ π₯=15/2 is local minima Thus, S(π₯) is Minimum at π₯=15/2 Hence, 1st number = π₯=ππ/π 2nd number = 15βπ₯=15β15/2=ππ/π