Let `f: RvecR` be a function satisfying condition `f(x+y^3)=f(x)+[f(y)]^3 for all x ,y in Rdot` If `f^(prime)(0)geq0,` find `f(10)dot`

Explanation

`"Given "f(x+y^(3))=f(x)+[f(y)]^(3)" (1)"`
`"and "f'(0)ge0" (2)"`
Replacing x, y by 0, we get
`f(0)=f(0)+f(0)^(3)orf(0)=0" (3)"`
`"Also, "f'(0)=underset(hrarr0)lim(f(0+h)-f(0))/(h)=underset(hrarr0)lim(f(h))/(h)" (4)"`
`"Let "I=f'(0)=underset(hrarr0)lim(f(0 +(h^(1//3))^(3))-f(0))/((h^(1//3))^(3))`
`=underset(hrarr0)lim(f((h^(1//3)))^(3))/((h^(1//3))^(3))=underset(hrarr0)lim((f(h^(1//3)))/((h^(1//3))))^(3)=I^(3)`
`"or "I=I^(3)`
`"or "I=0, 1,-as f'(0)ge0`
`therefore" "f'(0)=0,1`
`"Thus "f'(x)=underset(hrarr0)lim(f(x+h)-f(x))/(h)=underset(hrarr0)lim(f(x+(h^(1//3))^(3))-f(x))/((h^(1//3))^(3))`
`=underset(hrarr0)lim(f(x)+(f(h^(1//3)))^(3)-f(x))/((h^(1//3))^(3))" [using (1)]"`
`=underset(hrarr0)lim(f(h^(1//3))/((h^(1//3))))^(3)=(f'(0))^(3)`
`=0,1" [As f'(0)=0, 1 using (5)]"`
Integrating both sides, we get
`f(x)=c or x +c`
`"or "f(x)=0 or x (becausef(0) = 0)`
Thus, f(10) = 0 or 10