Ang f (x) = e ^ x / x-x ^ 3-3 concave o convex sa x = -1?

Sagot:

# Convex #

Paliwanag:

Upang suriin kung ang function ay convex o malukong mayroon kaming upang mahanap#f '' (x) #

Kung #color (brown) (f '' (x)> 0) # pagkatapos #color (brown) (f (x)) # ay #color (brown) (matambok) #

Kung #color (brown) (f '' (x) <0) # pagkatapos #color (brown) (f (x)) # ay #color (brown) (malukot) #

hanapin muna natin #color (asul) (f '(x)) #

#f '(x) = ((e ^ x) / x)' - (x ^ 3) '- (3)' #

#f '(x) = (xe ^ x-e ^ x) / x ^ 2-3x ^ 2-0 #

#color (asul) (f '(x) = (xe ^ x-e ^ x) / x ^ 2-3x ^ 2) #

Ngayon, hanapin natin #color (pula) (f '' (x)) #

(x) ^ x-e ^ x) 'x ^ 2 (x ^ 2)' (xe ^ x-e ^ x)) / (x ^ 2) ^ 2-6x #

#f '' (x) = ((e ^ x + xe ^ x-e ^ x) x ^ 2-2x (xe ^ x-e ^ x)) / x ^ 4-6x #

#f '' (x) = (x ^ 3e ^ x-2x ^ 2e ^ x-2xe ^ x) / x ^ 4-6x #

Let us simplify the fraction by # x #

#color (pula) (f '' (x) = (x ^ 2e ^ x-2xe ^ x-2e ^ x) / x ^ 3-6x) #

Ngayon ipaalam sa amin compute #color (brown) (f '' (- 1) #

#f '' (- 1) = ((- 1) ^ 2e ^ (- 1) -2 (-1) e ^ (- 1) -2e ^ (- 1)) / (- 1) ^ 3-6 (-1) #

#f '' (- 1) = (e ^ (- 1) + 2e ^ (- 1) -2e ^ (- 1)) / (- 1) + 6 #

#color (brown) (f '' (- 1) = - e ^ (- 1) +6) #

#color (brown) (f '' (- 1)> 0 #

Kaya,#f '' (x)> 0 # sa # x = -1 #

Samakatuwid,#f (x) # ay covex sa # x = -1 #

graph {e ^ x / x - x ^ 3 -3 -20, 20, -20, 20}