Example 30 - Find all points of local maxima, minima - CBSE - Examples

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise

Transcript

Example 30 (Method 1 ) Find all the points of local maxima and local minima of the unction f given by f ﷐𝑥﷯=2𝑥3 –6𝑥2+6𝑥+5. f (𝑥)=2𝑥3 –6𝑥2+ 6𝑥+5 Step 1: f’ (𝑥) f ′(𝑥)= ﷐𝑑﷐2𝑥3 – 6𝑥2+ 6𝑥 + 5﷯﷮𝑑𝑥﷯ f ′(𝑥)=6𝑥2 –12𝑥+6+0 f ′(𝑥)=6﷐﷐𝑥﷮2﷯−2𝑥+1﷯ Step 2: Putting f ′(𝑥)= 0 6﷐﷐𝑥﷮2﷯−2𝑥+1﷯=0 ﷐𝑥﷮2﷯−2𝑥+1=0 ﷐﷐𝑥﷯﷮2﷯+﷐﷐1﷯﷮2﷯−2﷐𝑥﷯﷐1﷯=0 ﷐﷐𝑥−1﷯﷮2﷯=0 ﷐𝑥−1﷯﷐𝑥−1﷯=0 So, 𝑥=1 is only critical point Step 3: Hence 𝒙=𝟏 is point of inflexion Example 30 (Method 2) Find all the points of local maxima and local minima of the unction f given by f ﷐𝑥﷯=2𝑥3 –6𝑥2+6𝑥+5. f (𝑥)=2𝑥3 –6𝑥2+ 6𝑥+5 Step 1: Finding f’ (𝑥) f ′(𝑥)= ﷐𝑑﷐2𝑥3 – 6𝑥2+ 6𝑥 + 5﷯﷮𝑑𝑥﷯ f ′(𝑥)=6𝑥3 –12𝑥+6+0 f ′(𝑥)=6﷐﷐𝑥﷮2﷯−2𝑥+1﷯ Step 2: Putting f ′(𝑥)= 0 6﷐﷐𝑥﷮2﷯−2𝑥+1﷯=0 ﷐𝑥﷮2﷯−2𝑥+1=0 ﷐﷐𝑥﷯﷮2﷯+﷐﷐1﷯﷮2﷯−2﷐𝑥﷯﷐1﷯=0 ﷐﷐𝑥−1﷯﷮2﷯=0 ﷐𝑥−1﷯﷐𝑥−1﷯=0 So, 𝑥=1 is only critical point Step 3: Finding f’’﷐𝑥﷯ f’﷐𝑥﷯=6﷐﷐𝑥﷮2﷯−2𝑥+1﷯ f’’﷐𝑥﷯=﷐6𝑑﷐﷐𝑥﷮2﷯ − 2𝑥 + 1﷯﷮𝑑𝑥﷯ f’’﷐𝑥﷯=6﷐2𝑥−2+0﷯ f’’﷐𝑥﷯=12﷐𝑥−1﷯ Putting 𝑥=1 f’’﷐1﷯=12﷐1−1﷯ f’’﷐𝑥﷯=12﷐𝑥−1﷯ At x = 1, f’’﷐𝑥﷯ =12﷐0﷯= 0 Thus, second derivative test fail Hence, 𝑥=1 is neither point of Maxima nor point of Minima ∴ 𝒙=𝟏 is point of inflexion.

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