Slide49.JPG

Slide50.JPG
Slide51.JPG Slide52.JPG Slide53.JPG Slide54.JPG Slide55.JPG Slide56.JPG Slide57.JPG

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Ex 6.3, 14 (Method 1) Find two positive numbers š‘„ and y such that š‘„ + š‘¦ = 60 and š‘„š‘¦3 is maximum. Given two number š‘„ and y, such that š‘„ + š‘¦ = 60 š‘¦=60āˆ’š‘„ Let P = š‘„š‘¦3 We need to maximize P Now, P = š‘„š‘¦3 Putting value of y from (1) P = š‘„(60āˆ’š‘„)3 Finding Pā€™(x) P = š‘„(60āˆ’š‘„)^3 Diff w.r.t š‘„ š‘‘š‘ƒ/š‘‘š‘„=š‘‘(š‘„(60 āˆ’ š‘„)^3 )/š‘‘š‘„ š‘‘š‘ƒ/š‘‘š‘„=š‘‘(š‘„)/š‘‘š‘„ (60āˆ’š‘„)^3+(š‘‘(60 āˆ’ š‘„)^3)/š‘‘š‘„ . š‘„ =(60āˆ’š‘„)^3+怖3(60āˆ’š‘„)怗^2 . (0āˆ’1) . š‘„ =(60āˆ’š‘„)^3āˆ’3š‘„(60āˆ’š‘„)^2 =(60āˆ’š‘„)^2 (60āˆ’š‘„)āˆ’3š‘„(60āˆ’š‘„)^2 =(60āˆ’š‘„)^2 [(60āˆ’š‘„)āˆ’3š‘„] =(60āˆ’š‘„)^2 [60āˆ’4š‘„] Putting š’…š‘·/š’…š’™=šŸŽ (60āˆ’š‘„)^2 (60āˆ’4š‘„)=0 So, x = 60 & x = 60/4 = 15 But, If š‘„=60, š‘¦= 60 ā€“ š‘„ = 60 ā€“ 60 = 0 Which is not possible Hence, š‘„= 15 is only critical point. Finding Pā€™ā€™ (š’™) Pā€™ā€™ (š‘„)=š‘‘((60 āˆ’ š‘„)^2 (60 āˆ’ 4š‘„))/š‘‘š‘„ Pā€™ā€™ (š‘„)=(š‘‘(60 āˆ’ š‘„)^2)/š‘‘š‘„ . (60āˆ’4š‘„)+š‘‘(60 āˆ’ 4š‘„)/š‘‘š‘„ (60āˆ’š‘„)^2 = 2(60āˆ’š‘„) .(0āˆ’1)(60āˆ’4š‘„)āˆ’4(60āˆ’š‘„)^2 = āˆ’2(60āˆ’š‘„) . (60āˆ’4š‘„)āˆ’4(60āˆ’š‘„)^2 = āˆ’2(60āˆ’š‘„)[(60āˆ’4š‘„)+2(60āˆ’š‘„)] = āˆ’2(60āˆ’š‘„)[(60āˆ’4š‘„)+120āˆ’2š‘„] = āˆ’2(60āˆ’š‘„)(180āˆ’6š‘„) At š’™ = 15 Pā€™ā€™(15)=āˆ’2(60āˆ’15)(180āˆ’6(15)) =āˆ’90 Ɨ90 =āˆ’8100 < 0 āˆ“ Pā€™ā€™(š‘„)<0 at š‘„ = 15 Hence š‘„š‘¦3 is Maximum when š‘„ = 15 Thus, when š‘„ = 15 š‘¦ =60 ā€“ š‘„=60 āˆ’15=45 Hence, numbers are 15 & 45

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.