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Ex 6.5,3 - Chapter 6 Class 12 Application of Derivatives - Part 6

Ex 6.5,3 - Chapter 6 Class 12 Application of Derivatives - Part 7
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Ex 6.5, 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) ℎ(𝑥)=sin⁡𝑥+cos⁡𝑥, 0<𝑥<𝜋/2 ℎ(𝑥)=sin⁡𝑥+cos⁡𝑥, 0<𝑥<𝜋/2 Finding 𝒉′(𝒙) ℎ′(𝑥)=𝑑(sin⁡𝑥 + cos⁡𝑥" " )/𝑑𝑥 ℎ^′ (𝑥)=cos⁡𝑥−sin⁡𝑥 Putting 𝒉′(𝒙)=𝟎 cos⁡𝑥−𝑠𝑖𝑛𝑥=0 cos⁡〖𝑥=𝑠𝑖𝑛 𝑥〗 1 = sin⁡𝑥/cos⁡𝑥 1 = tan 𝑥 tan 𝑥=1 ∴ 𝑥=45°= 𝜋/4 Finding h’’(𝒙) h’(𝑥)=cos⁡𝑥−sin⁡𝑥 h’’(𝑥)=−sin⁡𝑥−cos⁡𝑥 Putting 𝒙=𝝅/𝟒 h’’(π/4)=−sin(π/4)−𝑐𝑜𝑠(π/4) = – 1/√2−1/√2 = (−2)/√2 = – √2 Since h’’(𝑥)<0 when 𝑥=π/4 ∴ 𝑥=π/4 is point of Local Maxima f has Maximum value at 𝒙=𝝅/𝟒 f(𝑥)=𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥 f(π/4)=𝑠𝑖𝑛(π/4)+𝑐𝑜𝑠(π/4) = 1/√2+1/√2 = 2/√2 = √𝟐

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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.