Question 4:
If $f^{''} (x) = 8 \sin (4 x); f^{'} (2 π) = 1$ , and $f^{(0)} = 2$ , what function satisfy the given conditions?
A) $f (x) = \sin (4 x) + 3 x$
B) $f (x) = - \sin (4 x) + 3 x$
C) $f (x) = - \sin (4 x) - 3 x$
D) None of them

Answer

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Answer

f(x)= -2\int\cos(4x) dx + \int C dx = -\frac{1}{2}\sin(4x) + Cx + D

Steps

Step 1 ：\int 8\sin(4x) dx = 8 \int \sin(4x) dx

Step 2 ：f'(x)= 8 \int \sin(4x) dx = -2\cos(4x) + C

Step 3 ：\int -2\cos(4x) + C dx = -2\int\cos(4x) dx + \int C dx

Step 4 ：f(x)= -2\int\cos(4x) dx + \int C dx = -\frac{1}{2}\sin(4x) + Cx + D