3.2 Class Notes

# 3.2 Class Notes - Problem(a Find all the critical points(x...

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Problem (a) Find all the critical points (x values), (b) Use the second derivative test to determine if they are local max or min Find the open intervals on which the function is (c) increasing, (d) convex 4 2 ( ) 8 f x x x = Solution 4 2 3 2 2 2 ( ) 8 '( ) 4 16 4 ( 4) 4( 2) ( 2) ''( ) 12 16 0 convex when 4/3 crit. pts -2 0 2 min max min increasing 2 0, 2 f x x x f x x x x x x x x f x x x x x = = = = + = > > < < > f(x) = x 4 – 8x 2 crit. pts. -2,0,2 -20 -10 0 10 20 30 40 50 60 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Max-min word problem A parcel delivery service will only deliver a package if its length plus its girth (2w+2h) is less than 108 inches. What is the box of largest volume with w=h that they will deliver? Max-min word problem A parcel delivery service will only deliver a package if its length plus its girth (2w+2h) is less than 108 inches. What is the box of largest volume with w=h that they will deliver? 108 = L + 2w + 2h = L + 4w Maximize Lwh = (108-4w) w 2 Maximize (108-4w) w 2 L = 108 - 4w 2 3 2 ( ) 108 4 '( ) 216 12 0 when 0,18 ''( ) 216 24 convex when 9 ''(18) 0 max ''(0) 216 min '( ) 12 (18 ) 0 18 increasing 0

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