What are the points of inflection, if any, of #f(x)= x^5+ x^3- x^2-2 #?

Calculate the first and second derivatives

#f(x)=x^5+x^3-x^2-2#

#f'(x)=5x^4+3x^2-2x#

#f''(x)=20x^3+6x-2#

The points of inflection are when

#f''(x)=0#

#20x^3+6x-2=0#

#<=>#, #10x^3+3x-1=0#

The solution of this equation is obtained graphically

#x=0.269#

The point of inflection is #=(0.269, -2.051)#

graph{(y-(x^5+x^3-x^2-2))(y-(10x^3+6x-2))=0 [-2.712, 4.217, -4.075, -0.61]}