Example 17 - Chapter 6 Class 12 Application of Derivatives
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 17 (Method 1) Find all points of local maxima and local minima of the function f given by 𝑓 (𝑥)=𝑥3 – 3𝑥 + 3.Finding maximum & minimum value of 𝑓(𝑥)=𝑥3 – 3𝑥+3 Minimum value 𝑓(1)=(1)3 –3(1)+3= 1 – 3 + 3 = 1 Maximum value 𝑓(−1)=(−1)3 –3(−1)+3= –1 +3 + 3 = 5 Example 17 (Method 2) Find all points of local maxima and local minima of the function f given by 𝑓 (𝑥) = 𝑥3 – 3𝑥 + 3. 𝑓(𝑥)=𝑥3 – 3𝑥+3 Finding 𝒇^′ (𝒙) 𝑓′(𝑥)= 3𝑥^2 – 3+0 𝑓′(𝑥)= 3(𝑥^2−1) Putting 𝒇′(𝒙)= 0 3(𝑥^2−1)=0 𝑥^2−1=0 (𝑥−1)(𝑥+1)=0 So, x = 1 & x = −1 Finding 𝒇′′(𝒙) 𝑓^′ (𝑥)=3(𝑥^2−1) 𝑓^′′ (𝑥)=3 𝑑(𝑥^2 − 1)/𝑑𝑥 𝑓^′′ (𝑥)=3(2𝑥−0) 𝑓^′′ (𝑥)=6𝑥 Putting 𝒙=−𝟏 𝑓^′′ (𝑥) = 6(−1) <0 𝑥 = –1 is point of local maxima Putting 𝒙=𝟏 𝑓^′′ (𝑥) = 6(1) >0 𝑥 = 1 is point of local minima Finding maximum & minimum value of 𝑓(𝑥)=𝑥3 – 3𝑥+3 Minimum value at x = 1 𝑓(1)=(1)3 –3(1)+3= 1 – 3 + 3 = 1 Maximum value at x = −1 𝑓(−1)=(−1)3 –3(−1)+3= –1 + 3 + 3 = 5
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About the Author
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