Question

4. What are the absolute extrema (if any) coordinates for the function in #6? 5. What are the absolute extrema coordinates for the function, f(x) = x + 2x2 - 4x + 1, on the interval (-3, 1)? (x + 2x - 1)(x + 3) 6. Find the derivative of the following function.f(x) x+2 7. Suppose you have a rectangle piece of cardboard that is 16 inches by 7 inches that you want to use to make a box that is open at the top. Squares of equal sides x will be cut out of each corner then the sides are folded upwards to make the box. For what value of x will maximize the volume of the box? [Hint: draw a picture and imagine cutting out x by x squares in each comer. What would that make the length, width and height of the box?)

Answer 1

Sol: f(-) = (a +3*-4) (x+3) (x+2) desiratine of f(x) ds (x3 + 2x-6) (x+3) } da ? (x+2) using porsdech dgx422 - 4x +32 ² 4 6x + } de 2 ds at 4 32 ² + 2x² + 2x - 12 using quotieht sule of desirative (2+2) d ( 4 32 422 428-12) (443x?+ 221 422 -12) dae ? dhe d dac (x+2) て (2+) = (2+2) (42° 49x +48 +2) - (3*3*+ 2x73x=12270 (x+2) ? tx'19*44x72x+3x+ +18°782 +Y. - XY_323_27²_2242 (x+22

= 3x + 1403 +20x² + 8x +16 (x + 2)2 f'(o) - 3x4+147° +2 08 + 838 +16 (x+2) 2 For absolute extrema Co osdinats f (x) 3344147 +20% +82776 (x+2) 304 +1482? +20x' +88+16=0 For the above equation There 'n' where equation become No Critical value Zero No absolute extrema points us

Sola: 1-3,1 f(x) = x²+2x² 4x+1 finding (sitical value for O det da x² + 2002-400 +1} : 301 +42€ - 4o 3x² + 6x-2x_4=0 3x(x+2)-2(x+2) = 0 (&-f2) (3x-2) =0 X = 2 & x= 2% i Caitical point x = -2, 3 So, now, 4 9 &(-2): f(243) - -13737 absolute Maxima Co-ordinate= (-2, 9) absolute Minima Co ordinate: (/3, -13/1) g