[微分] 微分的證明題

[caxz (虛幻夢影)]

(1)
f(x),f'(x) and f''(x) are all continous on I
Suppose x=a is a critical point of f(x)
show that

(i)If f''(x)≧0 for all x屬於I
   then x=a is a global minimizer of f(x)

(ii) If f''(x)>0
     then x=a is a strict local minimizer of f(x)

(提示  用泰勒展開證明之)




(2)
Show that any continous function f:[0,1]->[0,1] has at least one fixed point
(i.e. 存在c屬於(0,1)-> f(c)=c  )



(3)
         { 1 , x>0 y>0
f(x,y) = { 0 , o.w.

show that f_x and f_y both existat(0,0) but it is not differentiable at (0,0)



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