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Last updated at Jan. 7, 2020 by Teachoo

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Example 41 An Apache helicopter of enemy is flying along the curve given by ๐ฆ= ๐ฅ^2 + 7. A soldier, placed at (3, 7), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance. The curve is given as y = x2 + 7 Let helicopter be at point (x, y) = (x, x2 + 7) Let d be the distance between helicopter soldier at (3, 7) d = โ((๐ฅ2โ๐ฅ1)2+(๐ฆ2โ๐ฆ1)2) d = โ((๐ฅโ3)2+(ใ(๐ฅใ^2+7)โ7)2) d = โ((๐ฅโ3)2+(๐ฅ)4) We need to find nearest distance i.e. minimum value of d Let f(x) = d2 f(x) = (x โ 3)2 + x4 When f(x) is minimum, d is minimum Finding fโ(x) fโ(x) = 2(x โ 3) + 4x3 = 2x โ 6 + 4x3 = 4x3 โ 2x โ 6 (To make calculation easy) Factorizing fโ(x) fโ(1) = 4(1)3 โ 2(1) โ 6 = 4 + 2 โ 6 = 0 Hence, (x โ 1) is a factor of 4x3 โ 2x โ 6 Thus, fโ(x) = (x โ 1) (4x2 + 4x + 6) Hence fโ (x) = 0 gives x โ 1 = 0 x = 1 2x2 + 2x + 3 = 0 x = (โ2 ยฑ โ(4 โ 4(2)(3)))/4 x = (โ2 ยฑ โ(4 โ 24))/4 x = (โ2 ยฑ โ(โ20))/4 This is not possible as there are no real roots. Hence there is only one point x = 1 This is either the maxima or minima. Hence we find fโ(x) fโโ (x) = (4x3 + 2x โ 6)โ fโโ(x) = 12x2 + 2 Finding value at x = 1,. fโโ(1) = 12(1)2 + 2 = 12 + 2 = 14 Since fโโ(x) > 0 โด x = 1 is the minima. The value of f(1) is f(1) = (1 โ 3)2 + 14 = (โ2)2 + 1 = 4 + 1 = 5 Hence, minimum distance between soldier & Helicopter d = โ(๐(1)) d = โ๐

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Chapter 6 Class 12 Application of Derivatives

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.