Question

1. Solve the following a. Sketch the graph of f (x) = 3x4 – x3 showing all the following key features i. Relative maximums and minimums ii. Intervals of increase and decrease iii. Points of inflection b. A business owner estimates that if x units of a particular commodity are produced, the total cost will be C(x) dollars where C(x) = 2x3 – 18x2 + 350x. At what level of production will the average cost A(x) = CVPbe minimized?

Answer 1

أو ته دا OA -0.002 -0.000 O) Given fex) = 3x4 – x3 for Gritical point find flx) |'(x)=12*3 - 3x²=0 3x2(4x-1)=0 X=0, x=”. find f (x) = 36x² - 6x cat x = f , f g = 9 - - 2 2 3 3 ) Bo local minina at x = ď There will be No any relative maxima and Relative minima at x = /4 = 0.25 So Relative minima (x, f(x1 = (0.25, 0.6639) lis for increasing decreasing if f'(x) >0 then increasing spot ↑ other gift f'(x) <0 then decreesing 8, in creasin on interval = (-00,oo U (4 100)) and decreasing on interval = (0,4) Any in 55 ) Point of inflection when f(x) = 0 atx=0 so point of inflection at x =0 Aug.

i Point of infraction - When f(c) = 0 whor xoc is a critical point then there will point of inflection to point of inflection at x=0 Any b) Given Cost C(X) = 2x3_18x2+350x for are. Cost A(x) = ct = 2x2_18*4350 for maximize AG find A'(X) = 4X-18=2 x = 4.5 at x=4.5 All = 2x14.532 – 18x4.5 +350 A(x) = $309.5- so at level x=4.5 Ang average cost will be maximizol
solved in sheet

I Given, function f(x) = 4x4_ x3 197) Lt f (x) for critical point findt (x) {(x) = 16x3_ 3x² - x? ( 16x – 3) - - find double derivative $"(x) = 48x? – 6X at x = 1/16 f (3) >o 8o 80 local minima at a=3 local minima atr-3 16 a) i) There will No any relative maxima J An and Relative minima at x =3 - o.1975 & Relative minima (x, f(x) = (0.1875, -0.001648) Ary for increasing, decreasing B oo thic to in which aren then function will be increasing on this aren other wise ole creasing fonchon. 8o fonction will be increasin on interval - = (-00,0) U ( 32,00) dans and function will be decreasing on interval = 10,3) the

i Point of infraction - When f(c) = 0 whor xoc is a critical point then there will point of inflection to point of inflection at x=0 Any b) Given Cost C(X) = 2x3_18x2+350x for are. Cost A(x) = ct = 2x2_18*4350 for maximize AG find A'(X) = 4X-18=2 x = 4.5 at x=4.5 All = 2x14.532 – 18x4.5 +350 A(x) = $309.5- so at level x=4.5 Ang average cost will be maximizol