Slide12.JPG

Slide13.JPG
Slide14.JPG


Transcript

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

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

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, Social Science, Physics, Chemistry, Computer Science at Teachoo.