Question

Given that f"(x)=2-2/x3 and f'(1)=0,find f'(x).Given further that f(1)=8, find f(x).

Answer 1

Given that $f''(x)=2-\frac{2}{x^3}$ and $f'(1)=0$ , let us find $f'(x):$

$f'(x)=2x+\frac{1}{x^2}+C_1$

$0=f'(1)=2+1+C_1,$ and hence $C_1=-3$

$f'(x)=2x+\frac{1}{x^2}-3$

Given further that $f(1)=8$ , let us find $f(x):$

$f(x)=x^2-\frac{1}{x}-3x+C_2$

$8=f(1)=1-1-3+C_2$

$C_2=11$

We conclude that $f(x)=x^2-\frac{1}{x}-3x+11.$