find f if.....f''(t) = 5 / sqrt(t) , f(4)=18 and f'(4)=4?

Question

find the anti-derivatives of f''t until you get to f...thanks a bunch!

Answer 1

f''(t) = 5.t^(-1/2)
f'(t) = 5.t^(1/2) / (1/2) + c = 10 t^(1/2) + c
f'(4) = 10√4 + c = 4 => c = -16
So f'(t) = 10 t^(1/2) - 16
f(t) = 10 t^(3/2) / (3/2) - 16t + d
= (20/3) t^(3/2) - 16t + d
f(4) = (20/3) 4^(3/2) - 64 + d = 18
So d = 82 - 160/3 = 86/3.

So f(t) = (20/3) t^(3/2) - 16t + 86/3.