`f''(x) = (2/3)x^(2/3)` Find `f`.

Expert Answers
gsarora17 eNotes educator| Certified Educator

`f''(x)=2/3x^(2/3)`

`f'(x)=int2/3x^(2/3)dx` 

`f'(x)=2/3(x^(2/3+1)/(2/3+1))+c_1`

`f'(x)=(2/3)(3/5)x^(5/3)+c_1`

`f'(x)=2/5x^(5/3)+c_1` 

c_1 is constant

`f(x)=int(2/5x^(5/3)+c_1)dx`

`f(x)=int2/5x^(5/3)dx+intc_1dx`

`f(x)=2/5(x^(5/3+1)/(5/3+1))+c_1x+c_2`

`f(x)=(2/5)(3/8)x^(8/3)+c_1x+c_2`

c_1 and c_2 are constants and simplifying the above the function is

`f(x)=3/20x^(8/3)+c_1x+c_2`