Ex 6.3, 3 (v) - Find local Maxima and Minima for f(x) = x^3 - 6x^2 + 9 - Ex 6.3

part 2 - Ex 6.3, 3 (v) - Ex 6.3 - Serial order wise - Chapter 6 Class 12 Application of Derivatives
part 3 - Ex 6.3, 3 (v) - Ex 6.3 - Serial order wise - Chapter 6 Class 12 Application of Derivatives

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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: (v) 𝑓 (𝑥)=𝑥3 –6𝑥2+9𝑥+15Putting f’(𝒙)=𝟎 3(𝑥^2−4𝑥+3)=0 𝑥^2−4𝑥+3=0 𝑥^2−3𝑥−𝑥+3=0 𝑥(𝑥−3)−1(𝑥−3)=0 (𝑥−1)(𝑥−3)=0 So, x = 1 & x = 3 Finding f’’(𝒙) f’(𝑥)=3(𝑥^2−4𝑥+3) f’’(𝑥)=𝑑(3(𝑥^2 − 4𝑥+3))/𝑑𝑥 = 3(2𝑥−4+0) = 6𝑥−12 Putting 𝒙=𝟏 in f’’(𝒙) f’’(1)=6(1)−12 = 6 – 12 = – 6 < 0 Since f’’(𝑥)<0 when 𝑥=1 ⇒ 𝑥=1 is point of local maxima ∴ f(𝑥) is maximum at 𝒙=𝟏 Maximum value of f(𝑥) at 𝑥 = 1 f(𝑥)=𝑥^3−6𝑥^2+9𝑥+15 f(1)=(1)^3−6(1)^2+9(1)+15 = 1 – 6 + 9 + 15 = 19 Putting 𝒙=𝟑 in f’’(x) f’’(𝑥)=6𝑥−12 f’’(3)=6(3)−12 = 18 – 12 = 6 > 0 Since f’’(𝑥)>0 when 𝑥=3 ⇒ 𝑥=3 is point of local minima ∴ f(𝑥) is minimum at 𝒙=𝟑 Minimum value of f(𝑥) at 𝑥 = 3 f(𝑥)=𝑥^3−6𝑥^2+9𝑥+15 f(3)=(3)^3−6(3)^2+9(3)+15 = 27 – 54 + 27 + 15 = 15

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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