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Example 33 - Find all points of local maxima, minima - NCERT - Examples

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Example 33 (Method 1) Find all the points of local maxima and local minima of the function f given by 𝑓(đ‘„)=2đ‘„3 –6đ‘„2+6đ‘„+5. 𝑓(đ‘„)=2đ‘„3 –6đ‘„2+6đ‘„+5 Step 1: Finding fâ€™ï·đ‘„ï·Ż đ‘“â€™ï·đ‘„ï·Ż=ï·đ‘‘ï·2đ‘„3 − 6đ‘„2 + 6đ‘„ + 5ï·Żï·źđ‘‘đ‘„ï·Ż đ‘“â€™ï·đ‘„ï·Ż=6ï·đ‘„ï·ź2ï·Żâˆ’12đ‘„+6+0 đ‘“â€™ï·đ‘„ï·Ż=6ï·ï·đ‘„ï·ź2ï·Żâˆ’2đ‘„+1ï·Ż đ‘“â€™ï·đ‘„ï·Ż=6ï·ï·ï·đ‘„ï·Żï·ź2ï·Ż+﷐﷐1ï·Żï·ź2ï·Żâˆ’2ï·đ‘„ï·Żï·1ï·Żï·Ż đ‘“â€™ï·đ‘„ï·Ż=6ï·ï·đ‘„âˆ’1ï·Żï·ź2ï·Ż Step 2: Putting fâ€™ï·đ‘„ï·Ż=0 6ï·ï·đ‘„âˆ’1ï·Żï·ź2ï·Ż=0 ï·ï·đ‘„âˆ’1ï·Żï·ź2ï·Ż=0 đ‘„âˆ’1=0 So, đ‘„=1 Step 3: Hence đ‘„=1 is point of inflexion Example 33 (Method 2) Find all the points of local maxima and local minima of the function f given by 𝑓(đ‘„)=2đ‘„3 –6đ‘„2+6đ‘„+5. 𝑓(đ‘„)=2đ‘„3 –6đ‘„2+6đ‘„+5 Step 1: Finding fâ€™ï·đ‘„ï·Ż đ‘“â€™ï·đ‘„ï·Ż=ï·đ‘‘ï·2đ‘„3 − 6đ‘„2+ 6đ‘„ + 5 ï·Żï·źđ‘‘đ‘„ï·Ż đ‘“â€™ï·đ‘„ï·Ż=6ï·đ‘„ï·ź2ï·Żâˆ’12đ‘„+6++0 đ‘“â€™ï·đ‘„ï·Ż=6ï·ï·đ‘„ï·ź2ï·Żâˆ’2đ‘„+1ï·Ż đ‘“â€™ï·đ‘„ï·Ż=6ï·ï·ï·đ‘„ï·Żï·ź2ï·Ż+﷐﷐1ï·Żï·ź2ï·Żâˆ’2ï·đ‘„ï·Żï·1ï·Żï·Ż đ‘“â€™ï·đ‘„ï·Ż=6ï·ï·đ‘„âˆ’1ï·Żï·ź2ï·Ż Step 2: Putting fâ€™ï·đ‘„ï·Ż=0 6ï·ï·đ‘„âˆ’1ï·Żï·ź2ï·Ż=0 ï·ï·đ‘„âˆ’1ï·Żï·ź2ï·Ż=0 đ‘„âˆ’1=0 đ‘„=1 Step 3: Finding fâ€™â€™ï·đ‘„ï·Ż fâ€™ï·đ‘„ï·Ż=6ï·ï·đ‘„âˆ’1ï·Żï·ź2ï·Ż fâ€™â€™ï·đ‘„ï·Ż=6.ï·đ‘‘ï·ï·đ‘„âˆ’1ï·Żï·ź2ï·Żï·źđ‘‘đ‘„ï·Ż fâ€™â€™ï·đ‘„ï·Ż=6 × 2ï·đ‘„âˆ’1ï·Ż fâ€™â€™ï·đ‘„ï·Ż = 12 ï·đ‘„âˆ’1ï·Ż Putting đ‘„=1 fâ€™â€™ï·đ‘„ï·Ż=12﷐1−1ï·Ż = 0 ∎ Second derivative test fails ⇒ đ‘„=1 is neither point of maxima nor point of minima Thus, 𝒙=𝟏 is point of inflexion

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