5
y`=4(x-1)³=0
x-1=0
x=1
_ +
-------------------(1)---------------
min
ymin=y(1)=(1-1)^4=0
7
y`=2(x+2)(x-3)²+2(x-3)(x+2)²=(x+2)(x-3)(2x-6+2x+4)=(x+2)(x-3)(4x-2)=0
x+2=0⇒x=-2
x-3=0⇒x=3
4x-2=0⇒x=0,5
_ + _ +
---------------------(-2)--------(0,5)---------------(3)---------------
min max min
ymin=y(-2)=y(3)=(-2+2)²(-2-3)²=0
ymax=y(0,5)=2,5²*(-2,5)²=39,0625
1
y`=(3x²(2+x)-1*x³)/(2+x)²=(6x+3x³-x³)/(2+x)²=(6x+2x³)/(2+x)²=0
2x(3+x²)=0
x=0;3+x²>0 при любом х
_ +
--------------------------(0)------------------
min
ymin=y(0)=0/2=0
2
y`=(5x^4(5-x)+1*x^5)/(5-x)²=(25x^4-5x^5+x^5)/(5-x)²=(25x^4-4x^5)/(5-x)²=0
x^4*(25-4x)=0
x=0 x=6,25
+ + _
-------------------(0)--------(6,25)----------------
max
ymax=y(6,25)=-5^9/2^8
3
1)x<5<br>y=(5-x)(x-3)³
y`=-1*(x-3)³+3(x-3)²(5-x)=(x-3)²(-x+3+15-3x)=(x-3)²(18-2x)=0
x-3=0⇒x=3
18-2x=0⇒x=9
+ + _
------------------(3)------------(9)---------
x=9∉(-∞;5)
точек экстремума нет
2)x≥5
y=(x-5)(x-3)³
y`=1*(x-3)³+3(x-3)²(x-5) =(x-3)²(x-3+3x-15)=(x-3)²(4x-18)=0
x-3=0⇒x=3
4x-18=0⇒x=4,5
_ _ +
------------------(3)------------(4,5)---------
x=4,5∉[5;∞)
точек экстремума нет
4
y`=(2(x-1)(x+1)-1(x-1)²)/(x+1)²=(x-1)(2x+2-x+1)/(x+1)²=(x-1)(x+3)/(x+1)²=0
x-1=0⇒x=1
x+3=0⇒x=-3
+ _ +
------------------(-3)------------(1)---------
max min
ymax=y(-3)=-8*(-6)³=-8*(-216)=1728
ymin=y(1)=-4*(-2)³=-4*(-8)=32