Ex 6.2,6 - Chapter 6 Class 12 Application of Derivatives

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Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (a) 𝑥2 + 2𝑥 – 5 f (𝑥) = 𝑥2 + 2𝑥 – 5 Step 1: Calculating f’ (𝑥) f’ (𝑥) = 2𝑥 + 2 f’ ( 𝑥) = 2 (𝑥 + 1) Step 2: Putting f’ (𝑥) = 0 2 (𝑥 + 1) = 0 (𝑥 + 1) = 0 𝑥 = –1 Step 3: Plotting point on real line Point 𝑥 = – 1 divide the real line into 2 disjoint interval i.e. ﷐−𕔴 −1﷯ & ﷐−1 , 𕔴uc1﷯ Step 4: Hence f is decreasing for ﷐−𕔴 –1﷯ f is increasing for ﷐ –1 ,𕔴uc1﷯ Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (b) 10 – 6𝑥 – 2𝑥2 Step 1: Calculating f’ (𝑥) f ﷐𝑥﷯ = 10 – 6𝑥 – 2𝑥2 f’ ﷐𝑥﷯ = –6 – 4𝑥 f’ ﷐𝑥﷯ = 2( –3 – 2𝑥) Step 2: Putting f’ ﷐𝑥﷯ = 0 2﷐−3 −2𝑥﷯=0 –3 – 2𝑥 = 0 –2𝑥 = 3 𝑥 = ﷐−3﷮2﷯ Step 3: Plotting point on real line Point 𝑥 = ﷐−3﷮2﷯ divide the real line into 2 disjoint interval i.e. ﷐−𕔴 ﷐−3﷮2﷯﷯ & ﷐﷐−3﷮2﷯ , 𕔴uc1﷯ Step 4: Hence f is decreasing for 𝒙 > ﷐−𝟑﷮𝟐﷯ f is increasing for 𝒙 < ﷐−𝟑﷮𝟐﷯ Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (c) –2𝑥3 – 9𝑥2 – 12𝑥 + 1 f ﷐𝑥﷯ = –2𝑥3 – 9𝑥2 – 12𝑥 + 1 Step 1: Calculating f’﷐𝑥﷯ f’ ﷐𝑥﷯ = –6𝑥2 –18𝑥 – 12 + 0 f’ ﷐𝑥﷯ = –6﷐𝑥2+3𝑥+2﷯ f’ ﷐𝑥﷯ = –6﷐𝑥2+2𝑥+𝑥+2﷯ f’ ﷐𝑥﷯ = –6﷐𝑥﷐𝑥+2﷯+1(𝑥+2)﷯ f’ ﷐𝑥﷯ = –6﷐﷐𝑥+1﷯( 𝑥+2)﷯ f’ ﷐𝑥﷯ = –6﷐𝑥+1﷯ ﷐𝑥+2﷯ Step 2: Putting f’ ﷐𝑥﷯ = 0 f’ ﷐𝑥﷯ = 0 – 6 ﷐𝑥+1﷯ ﷐𝑥+2﷯ = 0 ﷐𝑥+1﷯ ﷐𝑥+2﷯ = 0 So, 𝑥 = –1 , –2 Step 3: Plotting value of 𝑥 on real line Point 𝑥 = –2 & –1 divide the real line into 3 disjoint intervals i.e. ﷐−𕔴 −2﷯ ﷐−2 , −1﷯ & ﷐−1 , 𕔴uc1﷯ Step 4: Hence, f is increasing for –2 < 𝒙 < –1 & decreasing for 𝒙 < –2 & 𝒙 > –1 Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (d) 6 – 9𝑥 – 𝑥2 f﷐𝑥﷯ = 6 – 9𝑥 – 𝑥2 Step 1: Calculating f’ ﷐𝑥﷯ f’ ﷐𝑥﷯ = –9 – 2𝑥 f’ ﷐𝑥﷯ = –﷐2𝑥+9﷯ Step 2: Putting f’﷐𝑥﷯ = 0 –﷐2𝑥+9﷯ = 0 2𝑥 + 9 = 0 2𝑥 = – 9 𝑥 = ﷐−9﷮2﷯ Step 3: Plotting point 𝑥 = ﷐−9﷮2﷯ on real line Point 𝑥 = ﷐−9﷮2﷯ divide the real line into disjoint interval i.e. ﷐−𕔴 ﷐−9﷮ 2﷯﷯ & ﷐﷐−9﷮2﷯ , 𕔴uc1﷯ Step 4: Hence f is Strictly Increasing for 𝒙 < ﷐−𝟗﷮𝟐﷯ & Strictly Decreasing for 𝒙 > ﷐−𝟗﷮𝟐﷯ Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (e) ﷐﷐𝑥 + 1﷯﷮3﷯ ﷐﷐𝑥 – 3﷯﷮3﷯ f﷐𝑥﷯ = ﷐𝑥+1﷯3 ﷐𝑥−3﷯3 Step 1: Calculating f’﷐𝑥﷯ f﷐𝑥﷯ = ﷐𝑥+1﷯3 ﷐𝑥−3﷯3 f’﷐𝑥﷯= ﷐[﷐﷐𝑥+1﷯﷮3﷯]﷮′﷯﷐﷐𝑥−3﷯﷮3﷯ +﷐﷐﷐𝑥−3﷯3﷯﷮′﷯﷐﷐𝑥+1﷯﷮3﷯ f’﷐𝑥﷯=3﷐𝑥+1﷯2﷐𝑥−3﷯3 + 3﷐𝑥−3﷯2﷐𝑥+1﷯3 f’﷐𝑥﷯=3﷐𝑥+1﷯2﷐𝑥−3﷯2 ﷐﷐𝑥−3﷯+ ﷐𝑥+1﷯﷯ f’﷐𝑥﷯=3﷐𝑥+1﷯2﷐𝑥−3﷯2 ﷐2𝑥−2﷯ f’﷐𝑥﷯= 6﷐𝑥+1﷯2 ﷐𝑥−3﷯2 ﷐𝑥−1﷯ Step 2: Putting f’﷐𝑥﷯=0 6﷐𝑥+1﷯2 ﷐𝑥−3﷯2 ﷐𝑥−1﷯ = 0 Hence, 𝑥 = –1 , 1 , & 3 Step 3: Plotting values of x on real line. The point x = –1 , 1 & 3 divide the line segment into 4 disjoint intervals i.e. ﷐−𕔴 −1﷯ ﷐−1 , 1﷯ ﷐1 , 3﷯ & ﷐3 , 𕔴uc1﷯ Note that:- f’﷐𝑥﷯ = 6 ﷐﷐𝑥+1﷯﷮2﷯ ﷐﷐𝑥−3﷯﷮2﷯ ﷐𝑥−1﷯ Step 4: Plotting values of x on real line. Hence, f is strictly increasing for 1 < 𝒙 < 3 & 𝒙 > 3 f is strictly decreasing for 𝒙 < –1 & −𝟏<𝒙< 1