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

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

Grade:12

2 Answers

Manish Nihalani
askIITians Faculty 13 Points
one year ago
Dear Student,
 
Please find the solution for your query below
 
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
25763 Points
one year ago
Hello
 
Manish's answer is incorrect.
 
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

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