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Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (a) ๐ฅ2 + 2๐ฅ โ 5 f (๐ฅ) = ๐ฅ2 + 2๐ฅ โ 5 Step 1: Calculating fโ (๐ฅ) fโ (๐ฅ) = 2๐ฅ + 2 fโ ( ๐ฅ) = 2 (๐ฅ + 1) Step 2: Putting fโ (๐ฅ) = 0 2 (๐ฅ + 1) = 0 (๐ฅ + 1) = 0 ๐ฅ = โ1 Step 3: Plotting point on real line Point ๐ฅ = โ 1 divide the real line into 2 disjoint interval i.e. (โโ , โ1) & (โ1 , โ) Step 4: Hence f is strictly decreasing for (โโ ," โ1" ) f is strictly increasing for ( "โ1" ,โ) Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (b) 10 โ 6๐ฅ โ 2๐ฅ2 Step 1: Calculating fโ (๐ฅ) f (๐ฅ) = 10 โ 6๐ฅ โ 2๐ฅ2 fโ (๐ฅ) = โ6 โ 4๐ฅ fโ (๐ฅ) = 2( โ3 โ 2๐ฅ) Step 2: Putting fโ (๐ฅ) = 0 2(โ3 โ2๐ฅ)=0 โ3 โ 2๐ฅ = 0 โ2๐ฅ = 3 ๐ฅ = (โ3)/2 Step 3: Plotting point on real line Point ๐ฅ = (โ3)/2 divide the real line into 2 disjoint interval i.e. (โโ , (โ3)/2) & ((โ3)/2 , โ) Step 4: Hence f is strictly decreasing for ๐ > (โ๐)/๐ f is strictly increasing for ๐ < (โ๐)/๐ Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (c) โ2๐ฅ3 โ 9๐ฅ2 โ 12๐ฅ + 1 f (๐ฅ) = โ2๐ฅ3 โ 9๐ฅ2 โ 12๐ฅ + 1 Step 1: Calculating fโ(๐ฅ) fโ (๐ฅ) = โ6๐ฅ2 โ18๐ฅ โ 12 + 0 fโ (๐ฅ) = โ6(๐ฅ2+3๐ฅ+2) fโ (๐ฅ) = โ6(๐ฅ2+2๐ฅ+๐ฅ+2) fโ (๐ฅ) = โ6(๐ฅ(๐ฅ+2)+1(๐ฅ+2)) fโ (๐ฅ) = โ6((๐ฅ+1)( ๐ฅ+2)) fโ (๐ฅ) = โ6(๐ฅ+1) (๐ฅ+2) Step 2: Putting fโ (๐ฅ) = 0 fโ (๐ฅ) = 0 โ 6 (๐ฅ+1) (๐ฅ+2) = 0 (๐ฅ+1) (๐ฅ+2) = 0 So, ๐ฅ = โ1 , โ2 Step 3: Plotting value of ๐ฅ on real line Point ๐ฅ = โ2 & โ1 divide the real line into 3 disjoint intervals i.e. (โโ , โ2) (โ2 , โ1) & (โ1 , โ) Step 4: Hence, f is strictly increasing for โ2 < ๐ < โ1 & strictly decreasing for ๐ < โ2 & ๐ > โ1 Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (d) 6 โ 9๐ฅ โ ๐ฅ2 f(๐ฅ) = 6 โ 9๐ฅ โ ๐ฅ2 Step 1: Calculating fโ (๐ฅ) fโ (๐ฅ) = โ9 โ 2๐ฅ fโ (๐ฅ) = โ(2๐ฅ+9) Step 2: Putting fโ(๐ฅ) = 0 โ(2๐ฅ+9) = 0 2๐ฅ + 9 = 0 2๐ฅ = โ 9 ๐ฅ = (โ9)/2 Step 3: Plotting point ๐ฅ = (โ9)/2 on real line Point ๐ฅ = (โ9)/2 divide the real line into disjoint interval i.e. (โโ , (โ9)/( 2)) & ((โ9)/2 , โ) Step 4: Hence f is Strictly Increasing for ๐ < (โ๐)/๐ & Strictly Decreasing for ๐ > (โ๐)/๐ Ex 6.2,6 Find the intervals in which the following functions are strictly increasing or decreasing: (e) (๐ฅ + 1)^3 (๐ฅ โ 3)^3 f(๐ฅ) = (๐ฅ+1)3 (๐ฅโ3)3 Step 1: Calculating fโ(๐ฅ) f(๐ฅ) = (๐ฅ+1)3 (๐ฅโ3)3 Using product rule in (๐ฅ+1)3 (๐ฅโ3)3 As (๐ข๐ฃ)^โฒ=๐ข^โฒ ๐ฃ+๐ฃ^โฒ ๐ข fโ(๐ฅ)= ใ[(๐ฅ+1)^3]ใ^โฒ (๐ฅโ3)^3 +[(๐ฅโ3)3]^โฒ (๐ฅ+1)^3 fโ(๐ฅ)=3(๐ฅ+1)2(๐ฅโ3)3 + 3(๐ฅโ3)2(๐ฅ+1)3 fโ(๐ฅ)=3(๐ฅ+1)2(๐ฅโ3)2 ((๐ฅโ3)+ (๐ฅ+1)) fโ(๐ฅ)=3(๐ฅ+1)2(๐ฅโ3)2 (2๐ฅโ2) fโ(๐ฅ)= 6(๐ฅ+1)2 (๐ฅโ3)2 (๐ฅโ1) Step 2: Putting fโ(๐ฅ)=0 6(๐ฅ+1)2 (๐ฅโ3)2 (๐ฅโ1) = 0 Hence, ๐ฅ = โ1 , 1 , & 3 Step 3: Plotting values of x on real line. The point x = โ1 , 1 & 3 divide the line segment into 4 disjoint intervals i.e. (โโ , โ1) (โ1 , 1) (1 , 3) & (3 , โ)Note that:- fโ(๐ฅ) = 6 (๐ฅ+1)^2 (๐ฅโ3)^2 (๐ฅโ1) Step 4: Plotting values of x on real line. Hence, f is strictly increasing for 1 < ๐ฅ < 3 & ๐ฅ > 3 i.e. (1, 3) and (3, โ) f is strictly decreasing for ๐ฅ < โ1 & โ1<๐ฅ< 1 i.e. (โ โ, โ1) and (โ 1, 1)
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