Maximum slope of the curve y = –x 3 + 3x 2 + 9x – 27 is:

(A) 0                   (B) 12

(C) 16                 (D) 32

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Question 15 Maximum slope of the curve y = –x3 + 3x2 + 9π‘₯ – 27 is: 0 (B) 12 (C) 16 (D) 32 Given y = βˆ’ π‘₯^3 + 3π‘₯^2 + 9π‘₯βˆ’ 27 Now, Slope of the curve =π’…π’š/𝒅𝒙 = βˆ’3π‘₯^2 + 6π‘₯+ 9 We need to find maximum slope Let’s assume π’ˆ(𝒙) = Slope Thus, we need to maximize 𝑔(π‘₯) Maximizing π’ˆ(𝒙) 𝑔(π‘₯) = βˆ’3π‘₯^2 + 6π‘₯+ 9 Finding π’ˆβ€™(𝒙) π’ˆ^β€² (𝒙)=βˆ’6π‘₯+6 =βˆ’πŸ”(π’™βˆ’πŸ) Putting π’ˆ^β€² (𝒙)=𝟎 βˆ’6 (π‘₯βˆ’1) = 0 π‘₯βˆ’1 = 0 𝒙 = 1 Finding sign of g”(𝒙) at x = 1 g”(π‘₯) = βˆ’6 < 0 Since g’’(x) < 0 at x = 1 ∴ g is maximum at x = 1 Finding Maximum Value of g(x) g(1) = "βˆ’3 " γ€–(1)γ€—^2 "+ 6 (1) + 9" = βˆ’3 + 6 + 9 = βˆ’3 + 15 = 12 Hence, Maximum slope = Maximum value of g(x) = 12 So, the correct answer is (B)

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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 and Computer Science at Teachoo