उत तर:
# f_ (न य नतम) = f (1/4 + 2 ^ (- 5/3)) = (2 ^ (2/3) + 3 + 2 ^ (5/3)) / 4. #
स पष ट करण:
उसक अवल कन कर, #F (x) = 4x ^ 2-2x + x / (एक स 1/4); आरआर म x- {1/4}। #
# = 4x ^ 2-2x + 1 / 4-1 / 4 + {(एक स 1/4) +1/4} / (एक स 1/4); xne1 / 4 #
# = (2x-1/2) ^ 2-1 / 4 + {(एक स 1/4) / (एक स 1/4) + (1/4) / (एक स 1/4)}; xne1 / 4 #
# = 4 (एक स 1/4) ^ 2-1 / 4 + {1 + (1/4) / (एक स 1/4)}; xne1 / 4 #
#:। f (x) = 4 (एक स-1/4) ^ 2 + 3/4 + (1/4) / (एक स 1/4); xne1 / 4. #
अब, क ल ए स थ न य एक स ट र म, #F '(x) = 0, # तथ, #f '' (x)> य <0, "क अन स र" f_ (न य नतम) य f_ (अध कतम), "ब द । # #
#F '(x) = 0 #
#rrr 4 {2 (x-1/4)} + 0 + 1/4 {(- 1) / (x-1/4) ^ 2} = 0 … (ast) #
#rrr 8 (x-1/4) = 1 / {4 (x-1/4) ^ 2}, य, (x-1/4) ^ 3 = 1/32 = 2 ^ -5 #।
# आरएआरआर एक स = 1/4 + 2 ^ (- 5/3) #
आग क, # (ast) rArr f '' (x) = 8-1 / 4 {-2 (x-1/4) ^ - 3}, "त क," #
#F '' (1/4 + 2 ^ (- 5/3)) = 8+ (1/2) (2 ^ (- 5/3)) ^ - 3> 0 #
# "इसल ए," f_ (न य नतम) = 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.#
इस प रक र, #f_ (न य नतम) = f (1/4 + 2 ^ (- 5/3)) = (2 ^ (2/3) + 3 + 2 ^ (5/3)) / 4. #
गण त क आन द ल ।
