Given f ' ' ( x ) = − 25 sin ( 5 x ) and f ' ( 0 ) = − 4 and f ( 0 ) = − 5 . Find f ( π 2 ) =

Question

Given  and  and .

Find 

Answer 1

As f''(x) = -25sin(5x), f(x) = sin(5x) + ax + b, a and b unknown.

f(0) =sin(5*0) + a*0 + b = sin(0) + 0 + b = 0 + 0 + b = b

f'(x) = 5cos(5x) + a, so f'(0) = 5 cos(0) + a = 5*1 + a = 5 + a

f(0) = -5, so b = -5

f'(0)= -4, so 5 + a = -4, and a = -9

f(x) = sin(5x) - 9x - 5