Ex 6.3, 3 Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (iii) β„Ž(π‘₯)=sin⁑π‘₯+cos⁑π‘₯, 0<π‘₯<πœ‹/2 β„Ž(π‘₯)=sin⁑π‘₯+cos⁑π‘₯, 0<π‘₯<πœ‹/2 Finding 𝒉′(𝒙) β„Žβ€²(π‘₯)=𝑑(sin⁑π‘₯ + cos⁑π‘₯" " )/𝑑π‘₯ β„Ž^β€² (π‘₯)=cos⁑π‘₯βˆ’sin⁑π‘₯ Putting 𝒉′(𝒙)=𝟎 cos⁑π‘₯βˆ’π‘ π‘–π‘›π‘₯=0 cos⁑〖π‘₯=𝑠𝑖𝑛 π‘₯γ€— 1 = sin⁑π‘₯/cos⁑π‘₯ 1 = tan π‘₯ tan π‘₯=1 ∴ π‘₯=45Β°= πœ‹/4 Finding h’’(𝒙) h’(π‘₯)=cos⁑π‘₯βˆ’sin⁑π‘₯ h’’(π‘₯)=βˆ’sin⁑π‘₯βˆ’cos⁑π‘₯ Putting 𝒙=𝝅/πŸ’ h’’(Ο€/4)=βˆ’sin(Ο€/4)βˆ’π‘π‘œπ‘ (Ο€/4) = – 1/√2βˆ’1/√2 = (βˆ’2)/√2 = – √2 Since h’’(π‘₯)<0 when π‘₯=Ο€/4 ∴ π‘₯=Ο€/4 is point of Local Maxima f has Maximum value at 𝒙=𝝅/πŸ’ f(π‘₯)=𝑠𝑖𝑛π‘₯+π‘π‘œπ‘ π‘₯ f(Ο€/4)=𝑠𝑖𝑛(Ο€/4)+π‘π‘œπ‘ (Ο€/4) = 1/√2+1/√2 = 2/√2 = √𝟐

Go Ad-free
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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.