





Ex 6.5
Ex 6.5, 1 (ii) You are here
Ex 6.5, 1 (iii) Important
Ex 6.5, 1 (iv)
Ex 6.5, 2 (i)
Ex 6.5, 2 (ii) Important
Ex 6.5, 2 (iii)
Ex 6.5, 2 (iv) Important
Ex 6.5, 2 (v) Important
Ex 6.5, 3 (i)
Ex 6.5, 3 (ii)
Ex 6.5, 3 (iii)
Ex 6.5, 3 (iv) Important
Ex 6.5, 3 (v)
Ex 6.5, 3 (vi)
Ex 6.5, 3 (vii) Important
Ex 6.5, 3 (viii)
Ex 6.5, 4 (i)
Ex 6.5, 4 (ii) Important
Ex 6.5, 4 (iii)
Ex 6.5, 5 (i)
Ex 6.5, 5 (ii)
Ex 6.5, 5 (iii) Important
Ex 6.5, 5 (iv)
Ex 6.5,6
Ex 6.5,7 Important
Ex 6.5,8
Ex 6.5,9 Important
Ex 6.5,10
Ex 6.5,11 Important
Ex 6.5,12 Important
Ex 6.5,13
Ex 6.5,14 Important
Ex 6.5,15 Important
Ex 6.5,16
Ex 6.5,17
Ex 6.5,18 Important
Ex 6.5,19 Important
Ex 6.5, 20 Important
Ex 6.5,21
Ex 6.5,22 Important
Ex 6.5,23 Important
Ex 6.5,24 Important
Ex 6.5,25 Important
Ex 6.5,26 Important
Ex 6.5, 27 (MCQ)
Ex 6.5,28 (MCQ) Important
Ex 6.5,29 (MCQ)
Last updated at Aug. 19, 2021 by Teachoo
Ex 6.5, 1 (Method 1) Find the maximum and minimum values, if any, of the following functions given by (ii) f (π₯) = 9π₯2+12π₯+2Finding fβ(x) f (π₯)=9π₯2+12π₯+2 Diff. w.r.t π₯ fβ(π₯)=18π₯+12 fβ(π₯)=6(3π₯+2) Putting fβ(π)=π 6(3π₯+2)=0 3π₯+2=0 3π₯=β2 π₯=(β2)/( 3) Hence π₯=(β2)/3 is point of minima of f(π₯) Finding minimum value of f(π₯) at π₯=(β2)/3 f(π₯)=9π₯^2+12π₯+2 Putting π₯=(β2)/3 f(π₯)=9((β2)/3)^2+12((β2)/3)+2=9(4/3)β12(2/3)+2=β2 Thus, Minimum value of f(π)=βπ There is no maximum value Ex 6.5, 1 (Method 2) Find the maximum and minimum values, if any, of the following functions given by (ii) f (π₯) = 9π₯2+12π₯+2Finding fβ(π) f (π₯)=9π₯2+12π₯+2 Diff. w.r.t π₯ fβ(π₯)=π(9π₯^2 + 12π₯ + 2)/ππ₯ fβ(π₯)=18π₯+12 fβ(π₯)=6(3π₯+2) Putting fβ(π)=π 6(3π₯+2)=0 3π₯+2=0 3π₯=β2 π₯=(β2)/3 Finding fββ(π) fβ(π₯)= 6(3π₯+2) Again diff w.r.t π₯ fββ(π₯)=π(6(3π₯+2))/ππ₯ fββ(π₯)=6 π(3π₯ + 2)/ππ₯ fββ(π₯)=6(3+0) fββ(π₯)=6(3) fββ(π₯)=18 So, fββ((β2)/3)=18 Since fββ(π₯)>0 is for π₯=(β2)/3 π₯=(β2)/3 is point of local minima Finding minimum value Putting π₯=(β2)/3 in f(π₯) f (π₯)=9π₯2+12π₯+2 f ((β2)/3)=9((β2)/3)^2+12((β2)/3)+2 =9(4/9)+12((β2)/3)+2 =4β8+2 =β2 Hence, minimum value = β2 There is no maximum value