find absolute extrema on interval: f(x)=(2x-1)^2(x+5), [0,1]

[KingAce]

I really need to know how to solve these two problems in preparation for a quiz tomorrow! Please help

1. f(x) = (2x - 1)^2 (x+5) [0,1]

2. f(x) = x/2 + sinx [0, 2pi]

Any work you could show would be greatly appreciated. Thanks a ton

 

[arthur ohlsten]

Re: Need help finding ABSOLUTE EXTREMA! Please help..

given f[x]= [2x-1]^2[x+5]
find max in region 0,1

f[x]= [4x^2-4x+1][x+5]
f[x]= 4x^3 -16x^2-19x+5
f ' [x] = 12x^2-32x-19
slope is negative in the region of interest
max at x=0 f[0]= 5

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f[x]= x/2 + sin x region[0,2pi]
f ' [x] = 1/2 + cos x set F ' =0
cos x=-1/2
x=120degrees or 240 degrees
x=2 pi/3 or 4pi/3
from a right triangle
sin x= +/- sqrt3/2
f[120]=[2pi/3] /2 + sin 2pi/3
f[120]= pi/3 + sqrt3 /2

what is f[2pi]?
f[2pi] = pi + 0

is pi > pi/3 + sqrt3/2
is 2pi/3 > sqrt 3 /2
is 4 pi /3 > sqrt3
is 16 pi^2/9 >3
is 1.7 pi^2> 3 yes

maximim at x=2pi
f[2pi] = pi

Arthur