Browse Questions

# $f'(x) = a \cos x + b \sin x ;\; f'(0)=4;\; f(0)=3;\; f \bigg(\large\frac{\pi}{2}\bigg)$$=5. Find the value of \;f(x)\;? (a)\;4 \tan x + 2 \cot x +1\qquad(b)\;4 \sin x + 2 \cos x +1\qquad(c)\;4 \cos x + 2 \sin x +1\qquad (d)\;\cos x + \sin x Can you answer this question? ## 1 Answer 0 votes 1) Given, f'(x) = a \cos x + b \sin x, If x=0 \rightarrow f'(0) =4 4=a \cos (0) + b \sin (0) a=4 -----(i) Integrating both sides, \int f'(x) = a \cos x + b \sin x \rightarrow f(x) = + a \sin x - b \cos x+c -----(ii) 2) x = 0 \rightarrow f(0) = 3 \rightarrow f(0) =+a(0) -b(1) +c (from equation ii) \rightarrow -b+c=3 3) x=\large\frac{\pi}{2} \rightarrow f(\large\frac{\pi}{2})$$=5$
$5 =+a(1) -b(0) +c \rightarrow 5 =+a(1) -b(0) +c$
From (i) $\;\rightarrow c = 1$ ------ (iii)
Applying this in $-b+c=3$ we get $b=-2$ -----(iv)
From equation (i),(ii),(iii) and (iv) we get, $f(x) = 4 \sin x + 2 \cos x +1$
edited Mar 26, 2014