If f''(x)=-4sin2x, f(0)=0 and f'(0)=2, find the value of f(pi/4).

Answer 1

f'(x) = 2cos 2x + c
f'(0) = 2 ----> 0 = 2cos 0 + c
2 = 2 + c
c = 0

so f'(x) = 2cos 2x

then
f(x) = sin 2x + k
f(0) = 0 ----> 0 = sin 0 + k
k = 0
so
f(x) = sin 2x
f(π/4) = sin 2π/4 = 1