Ex 6.3

Chapter 6 Class 12 Application of Derivatives
Serial order wise

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Ex 6.3, 3 Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (v) π (π₯)=π₯3 β6π₯2+9π₯+15π (π₯)=π₯3 β6π₯2+9π₯+15 Finding fβ(π) fβ(π₯)=π(π₯3 β 6π₯2 + 9π₯ + 15" " )/ππ₯ fβ(π₯)=3π₯^2β12π₯+9 fβ(π₯)=3(π₯^2β4π₯+3) Putting fβ(π)=π 3(π₯^2β4π₯+3)=0 π₯^2β4π₯+3=0 π₯^2β3π₯βπ₯+3=0 π₯(π₯β3)β1(π₯β3)=0 (π₯β1)(π₯β3)=0 So, x = 1 & x = 3 Finding fββ(π) fβ(π₯)=3(π₯^2β4π₯+3) fββ(π₯)=π(3(π₯^2 β 4π₯+3))/ππ₯ = 3(2π₯β4+0) = 6π₯β12 Putting π=π in fββ(π) fββ(1)=6(1)β12 = 6 β 12 = β 6 < 0 Since fββ(π₯)<0 when π₯=1 β π₯=1 is point of local maxima β΄ f(π₯) is maximum at π=π Maximum value of f(π₯) at π₯ = 1 f(π₯)=π₯^3β6π₯^2+9π₯+15 f(1)=(1)^3β6(1)^2+9(1)+15 = 1 β 6 + 9 + 15 = 19 Putting π=π in fββ(x) fββ(π₯)=6π₯β12 fββ(3)=6(3)β12 = 18 β 12 = 6 > 0 Since fββ(π₯)>0 when π₯=3 β π₯=3 is point of local minima β΄ f(π₯) is minimum at π=π Minimum value of f(π₯) at π₯ = 3 f(π₯)=π₯^3β6π₯^2+9π₯+15 f(3)=(3)^3β6(3)^2+9(3)+15 = 27 β 54 + 27 + 15 = 15