Slide69.JPG

Slide70.JPG
Slide71.JPG Slide72.JPG

  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise

Transcript

Ex 6.5, 16 Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.Let first number be ๐‘ฅ Now, First number + second number =16 ๐‘ฅ + second number = 16 second number = 16 โ€“ ๐‘ฅ Now, Sum of Cubes = (๐‘“๐‘–๐‘Ÿ๐‘ ๐‘ก ๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ )^3+(๐‘ ๐‘’๐‘๐‘œ๐‘›๐‘‘ ๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ )^3 Let S(๐‘ฅ) = ๐‘ฅ3 + (16โˆ’๐‘ฅ)^3 We Need to Find Minimum Value of s(๐‘ฅ) Finding Sโ€™(๐‘ฅ) Sโ€™(๐‘ฅ)= ๐‘‘(๐‘ฅ^3+ (16 โˆ’ ๐‘ฅ)^3 )/๐‘‘๐‘ฅ = 3๐‘ฅ2 + 3(16โˆ’๐‘ฅ)^2. (0โˆ’1) = 3๐‘ฅ2 + 3(16โˆ’๐‘ฅ)^2 (โˆ’1) = 3๐‘ฅ2 โ€“ 3((16)^2+(๐‘ฅ)^2โˆ’2(16)(๐‘ฅ)) = 3๐‘ฅ2 โ€“ 3(256+๐‘ฅ^2โˆ’32๐‘ฅ) = 3๐‘ฅ2 โ€“ 3(256)โˆ’3๐‘ฅ^2+3(32)๐‘ฅ = โ€“3(256โˆ’32๐‘ฅ) Putting Sโ€™(๐‘ฅ)=0 โ€“3(256โˆ’32๐‘ฅ)=0 256 โ€“ 32๐‘ฅ = 0 32๐‘ฅ = 256 ๐‘ฅ = 256/32 ๐‘ฅ = 8 Finding Sโ€™โ€™(๐‘ฅ) Sโ€™(๐‘ฅ)=โˆ’3(256โˆ’32๐‘ฅ) Sโ€™โ€™(๐‘ฅ)=๐‘‘(โˆ’3(256 โˆ’ 32๐‘ฅ))/๐‘‘๐‘ฅ = โ€“3 ๐‘‘(256 โˆ’ 32๐‘ฅ)/๐‘‘๐‘ฅ = โ€“3 [0โˆ’32] = 96 > 0 Since Sโ€™โ€™(๐‘ฅ)>0 for ๐‘ฅ = 8 ๐‘ฅ = 8 is point of local minima & S(๐‘ฅ) is minimum at ๐‘ฅ = 8 Hence, 1st number = x = 8 & 2nd number = 16 โ€“ x = 16 โ€“ 8 = 8

About the Author

Davneet Singh's photo - Teacher, Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.