# Example 41 - Chapter 6 Class 12 Application of Derivatives

Last updated at Jan. 7, 2020 by Teachoo

Last updated at Jan. 7, 2020 by Teachoo

Transcript

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.