find f if f''(x)=

Question

Find f if f′(x)=6x−[3/x^4],  x>0, and f(1)=3.

 

f(x)=_____

Answer 1

f'(x) = 6x - (3/x⁴) 

to find f(x), we must integrate f'(x)

 

Use the power rule: 

if f'(x) = xⁿ, then f(x) = xⁿ⁺¹/(n+1) + C

where C is  the constant of integration. 

 

∴f(x) = 6x²/2 + 3/3x³ + C

f(x)= 3x² + 1/x³ + C

 

to find C, we use the condition given, f(1) =3. 

 

3 = 3(1)² + 1/1³ + C

C = -1

 

f(x)= 3x² + 1/x³ -1