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wy07

y=e^x+4e^-2xfind the coordinate and nature of the stationary points. (ans: (ln 2,3) minimum point) y=x^2 ln x find all the points of inflexion. (ans: e^-2/3 , -3e^-3/2)

peaceboy

y=e^x+4e^-2x
dy/dx = e^x -8e^(-2x)
dy/dx=0
e^x -8e^(-2x)=0
e^x=8e^(-2x)
x ln e = ln 8 x^(-2x)
         = -2x ln e + ln 8
3x = ln 8
x= (1/3)(ln8)
  = ln 8^(1/3)
  = ln 2

when x = ln 2 , y =e^ln 2 +4e^-2ln 2
                          = 2 +4(1/4)
                          =3
dy/dx = e^x -8e^(-2x)
d^2y/dx^2 = e^x + 16e^(-2x)
when ,x=ln2 ,d^2y/dx^2 >0
therefore (ln 2,3) is a minimum pnt

2.y=x^2 ln x

dy/dx = x^2(1/x) + (2x) ln x
         = x +(2x) ln x
d^2y/dx^2 = 1 + 2x(1/x) + 2ln x
d^2y/dx^2 = 0
1 + 2x(1/x) + 2ln x=0
lnx = -3/2
x = e^(-3/2)

y=x^2 ln x
y= (e^(-3/2))^2 ln  e^(-3/2)
  = e^-3 (-3/2)

o.0 答案有一点出入