# if f ‘(x) = a sin x + b cos x and f ‘(0) = 4 , f(0) = 3 , f(pi/2) =5 find f(x)

Manish Nihalani
3 years ago
Dear Student,

$f'(x) = a sinx + bcosx$

Since, $f'(0) = 4 f'(0) = a (0) + b(1) = 4\Rightarrow b = 4$

$Integrating, f'(x) , we get f(x) = -acosx + 4sinx + c$

$Now, f(0) = 3$

$-a + c = 3$…...(1)

Also,  $f(\Pi /2) = 5$

$4+ c = 5\Rightarrow c = 1$

Subsituting c = 1, in equation (1)

$a = -2$

$Therefore, f(x) = -2 cosx + 4sinx +1$

Arun
25757 Points
3 years ago
Hello

Dear student

f'(0) = 4 = b
f(x) = - a cos x + b sinx + c
Now
f(0) = 3 = -a + c
f(π/2) = 5 = b + c
Hence c = 1
And a = -2

Hence f(x) = 2 cos x + 4 sin x +1