Hva er den lokale ekstremiteten av f (x) = 4x ^ 2-2x + x / (x-1/4)?

Hva er den lokale ekstremiteten av f (x) = 4x ^ 2-2x + x / (x-1/4)?
Anonim

Svar:

# f_ (min) = f (1/4 + 2 ^ (- 5/3)) = (2 ^ (2/3) + 3 + 2 ^ (5/3)) / 4. #

Forklaring:

Vær oppmerksom på at, #f (x) = 4x ^ 2-2x + x / (x-1/4); x i RR- {1/4}. #

# = 4x ^ 2-2x + 1 / 4-1 / 4 + {(x-1/4) +1/4} / (x-1/4); xne1 / 4 #

# = (2x-1/2) ^ 2-1 / 4 + {(x-1/4) / (x-1/4) + (1/4) / (x-1/4)}; xne1 / 4 #

# 4 = (x-1/4) ^ 2-1 / 4 + {1 + (1/4) / (x-1/4)}; xne1 / 4 #

#:. f (x) = 4 (x-1/4) ^ 2 + 3/4 + (1/4) / (x-1/4); xne1 / 4. #

Nå for Lokal ekstrem, #f '(x) = 0 # og, #f '' (x)> eller <0, "som" f_ (min) eller f_ (maks), "resp." #

#f '(x) = 0 #

#rArr 4 {2 (x-1/4)} + 0 + 1/4 {(- 1) / (x-1/4) ^ 2} = 0 …

#rArr 8 (x-1/4) = 1 / {4 (x-1/4) ^ 2, eller, (x-1/4) ^ 3 = 1/32 = 2 ^ -5.

# rArr x = 1/4 + 2 ^ (- 5/3) #

Lengre, # (ast) rArr f '' (x) = 8-1 / 4 {-2 (x-1/4) ^ - 3}, "slik at"

#f '' (1/4 + 2 ^ (- 5/3)) = 8 + (1/2) (2 ^ (- 5/3)) ^ - 3> 0 #

# "Derfor" f_ (min) = f (1/4 + 2 ^ (- 5/3)) #

#=4(2^(-5/3))^2+3/4+(1/4)/(2^(-5/3))=2^2*2^(-10/3)+3/4+2^(-2)*2^(5/3)#

#=1/2^(4/3)+3/2^2+1/2^(1/3)=(2^(2/3)+3+2^(5/3))/4.#

Og dermed, #f_ (min) = f (1/4 + 2 ^ (- 5/3)) = (2 ^ (2/3) + 3 + 2 ^ (5/3)) / 4. #

Nyt matematikk.!