Example 29 - Find all points of local maxima and local minima - Local maxima and minima

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  1. Chapter 6 Class 12 Application of Derivatives
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Example 29 (Method 1) 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 Step 1: Finding ﷐𝑓﷮′﷯(𝑥) 𝑓′﷐𝑥﷯= 3﷐𝑥﷮2﷯ – 3+0 𝑓′﷐𝑥﷯= 3﷐﷐𝑥﷮2﷯−1﷯ Step 2: Putting 𝑓′﷐𝑥﷯= 0 3﷐﷐𝑥﷮2﷯−1﷯=0 ﷐𝑥﷮2﷯−1=0 ﷐𝑥−1﷯﷐𝑥+1﷯=0 Thus 𝑥 = 1 , –1 are only critical points Step 3: Example 29(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 Step 1: Finding ﷐𝑓﷮′﷯(𝑥) 𝑓′﷐𝑥﷯= 3﷐𝑥﷮2﷯ – 3+0 𝑓′﷐𝑥﷯= 3﷐﷐𝑥﷮2﷯−1﷯ Step 2: Putting 𝑓′﷐𝑥﷯= 0 3﷐﷐𝑥﷮2﷯−1﷯=0 ﷐𝑥﷮2﷯−1=0 ﷐𝑥−1﷯﷐𝑥+1﷯=0 So, x = 1 & x = −1 Step 3: 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 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

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