[1] Finding stat.pts:
f(x)=10x5-24x3
f'(x)=50x4-71x2=0 for stat.pts
2x2(25x2-36)=0
2x2(5x-6)(5x+6)=0
x = 0, 6/5, -6/5
f''(x)=200x3-144x
When x=0, f'(0)=0, f''(0)=0 [Possible pt of horizontal inflexion, test concavity]
{x}____ -0.1__0__0.1
{f''(x)}_14.2__0__-14.2
Concavity changes,
(0,0) is a pt of horizontal inflexion
When x=-6/5, f'(-6/5)=0, f''(-6/5)=-864/5 < 0
Concave downwards,
(-6/5,10368/625) is a maximum stationary pt
When x=6/5, f'(6/5)=0, f''(6/5)=864/5 > 0
Concave upwards,
(6/5,-10368/625) is a minimum stationary pt
[2] Finding other points of inflexion:
f''(x)=200x3-144x=0 for inflexion pts
8x(25x2-18)=0
8x(5x-√18)(5x+√18)=0
x=0, (3√2)/5, (-3√2)/5
When x=0 has already been found to be a pt of horizontal inflexion
When x=(3√2)/5, f''((3√2)/5)=0, testing concavity
{x}____[(3√2)/5]-0.1__[(3√2)/5]__[(3√2)/5]+0.1
{f''(x)}___-23.90...________0________34.09...
Concavity changes,
((3√2)/5,-10.26....) is a pt of inflexion
When x=(-3√2)/5, f''((-3√2)/5)=0, testing concavity
{x}____[(-3√2)/5]-0.1__[(3√2)/5]__[(-3√2)/5]+0.1
{f''(x)}___-34.09...________0________23.90...
Concavity changes,
((-3√2)/5,10.26....) is a pt of inflexion
Fuck finally. What a homo question seriously! That question was longer than my dick ffs.