Question

第八题求解

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Answer 1

(1)
y=x^5-5x^4+5x^3+1
y'=5x^4-20x^3+15x^2
y'=0
5x^2 .(x^2-4x+3)=0
5x^2 .(x-3)(x-1)=0
x=0 or 3 or 1
y''=20x^3-60x^2+30x
y'(0) =0
y'(1)<0 (max)

y(0) =1
y(-1) = -1-5-5+1 =-10
y(1) =1-5+5+1=2
y(2) = 32-5(16)+5(8)+1= 32-80+40+1 =-7

y=x^5-5x^4+5x^3+1 ; x ∈ [-1,2]
max y = y(1) =2
min y = y(2)=-7

(3)
y=x+√(1-x) ; x ∈ [-5,1]

1-x ≥0
x ≤1

y'= 1- 1/[2√(1-x)]
y'=0
2√(1-x) -1 =0
1-x = 1/4
x= 3/4

y'(3/4+) <0
y'(3/4-) >0
x=3/4 (max)

y(-5)=-5+√(1+5)=-5+√6
y(1) =1+√(1-1) = 1
y(3/4) =3/4+√(1-3/4) = 3/4 +1/2 = 5/4

max y = 5/4
min y =-5+√6