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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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.