How Derivatives Affect the Shape of a Graph

Question

Consider the function 𝑓(𝑥)= 12𝑥^5 + 75x^4 - 120𝑥x^3 + 7.

𝑓(𝑥) has inflection points at (reading from left to right) x = D, E, and F

where D is   

and E is   

and F is   

Answer 1

There is only one inflection point, (0, 7)

Answer 2

𝑓(𝑥)= 12𝑥^5 + 75x^4 - 120𝑥x^3 + 7

f' = 60x^4 + 300x^3 -360x^2

f'' = 240x^3+900x^2-720x

f" = 60x(4x^2+15x-12) = 0

x=0, quadratic formula: (-15+-sqrt((-15)^2-4*4*(-12))/(2*4) = (-15+-sqrt(417))/8

Possible POI: x=(-15+sqrt(417))/8, x=0, x=(-15-sqrt(417))/8

[x~~ -4.43, x=0, x~~ 0.678]

Check it is actually POI or not:

f"(-5) = -3900 <0, f"(-1)=1380>0, f"(1)=420<0 , f"(5)=48900>0

Implies all of them are POI

hence x=(-15+sqrt(417))/8, x=0, x=(-15-sqrt(417))/8

[x~~ -4.43, x=0, x~~ 0.678]