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Example 32 - Find local maximum and local minimum values

Example 32 - Chapter 6 Class 12 Application of Derivatives - Part 2
Example 32 - Chapter 6 Class 12 Application of Derivatives - Part 3 Example 32 - Chapter 6 Class 12 Application of Derivatives - Part 4 Example 32 - Chapter 6 Class 12 Application of Derivatives - Part 5


Transcript

Example 32 Find local maximum and local minimum values of the function f given by f (π‘₯)=3π‘₯4 + 4π‘₯3 – 12π‘₯2 + 12f (π‘₯)=3π‘₯4 + 4π‘₯3 – 12π‘₯2 + 12 Finding f’ (𝒙) f’ (π‘₯)=𝑑(3π‘₯4 + 4π‘₯3 – 12π‘₯2 + 12)/𝑑π‘₯ f’ (π‘₯)=12π‘₯^3+12π‘₯^2 – 24π‘₯ "+ 0" f’ (π‘₯)=12(π‘₯^3+π‘₯^2βˆ’2π‘₯) f’ (π‘₯)=12π‘₯(π‘₯^2+π‘₯βˆ’2) f’ (π‘₯)=12π‘₯ (π‘₯^2+2π‘₯βˆ’π‘₯βˆ’2) f’ (π‘₯)=12π‘₯ (π‘₯(π‘₯+2)βˆ’1(π‘₯+2)) f’ (π‘₯)=πŸπŸπ’™ (π’™βˆ’πŸ)(𝒙+𝟐) Putting f’ (𝒙)=𝟎 12π‘₯ (π‘₯βˆ’1)(π‘₯+2)=0 π‘₯ (π‘₯βˆ’1)(π‘₯+2)=0 So, 𝒙=𝟎,π‘₯=𝟏,& π‘₯=βˆ’πŸ Finding f’’(𝒙) f ’(π‘₯)=12(π‘₯^3+π‘₯^2βˆ’2π‘₯) f ’’(π‘₯)=12𝑑(π‘₯^3 + π‘₯^2 βˆ’ 2π‘₯)/𝑑π‘₯ f ’’(π‘₯)=𝟏𝟐(πŸ‘π’™^𝟐+πŸπ’™βˆ’πŸ) At 𝒙=𝟎 f ’’(0)=12(3(0)^2+2(0)βˆ’2)= 32 (0+0 βˆ’2)= – 64 < 0 Since f’’(π‘₯)<0 at π‘₯=0 ∴ π‘₯ = 0 is point of local maxima Thus, f(π‘₯) is maximum at π‘₯=0 At 𝒙=𝟏 f’’(1)=12(3(1)^2+2(1)βˆ’2)= 12 (3+2βˆ’2) = 36 > 0 Since f’’(π‘₯)>0 at π‘₯=1 ∴ π‘₯ = 1 is point of local minima Thus, f(π‘₯) is minimum at π‘₯=1 At 𝒙=βˆ’πŸ f’’(βˆ’2)=12(3(βˆ’2)^2+2(βˆ’2)βˆ’2)= 12 (12βˆ’4βˆ’2)= 72 > 0 Since f’’(π‘₯)>0 at π‘₯=βˆ’2 ∴ π‘₯ = βˆ’2 is point of local minima Thus, f(π‘₯) is minimum at π‘₯=βˆ’2 Finding local minimum and maximum value f’ (π‘₯)=πŸπŸπ’™ (π’™βˆ’πŸ)(𝒙+𝟐) Local maximum value of f (π‘₯) at π‘₯=0 f (0)=3(0)4 + 4(0)3 – 12(0)2 + 12 = 0 + 0 – 0 + 12 = 12 Local minimum value of f (π‘₯) at π‘₯=1 f (1)=3(1)4 + 4(1)3 – 12(1)2 + 12 = 3 + 4 – 12 + 12 = 7 Local Minimum value of f (π‘₯) at π‘₯=βˆ’2 f (βˆ’2)=3(βˆ’2)4 + 4(βˆ’2)3 – 12(βˆ’2)2 + 12 = 48 – 32 – 48 + 12 = – 20

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.