Question

Let f be a function defined for every x, such that f'' = -f ,f(0)=0, f' (0) = 1, then f(x) is equal to

• A. tanx
• B. $e^x-1$
• C. sinx
• D. 2sinx

Answer 1

The correct option is C sinx

$\int f^{{}^{\prime \prime }}\left(x\right)f^{{}^{\prime }}\left(x\right)=-\int f\left(x\right)f^{{}^{\prime }}\left(x\right)$
$\left(f^{{}^{\prime }}\right(x){)}^2=f^2(x)+c$
$c=1$
$f^{{}^{\prime }}\left(x\right)=\sqrt{1-f^2\left(x\right)}$
$\int \frac{df\left(x\right)}{\sqrt{1-f^2\left(x\right)}}=\int dx$
$si{n}^{-1}f\left(x\right)=x+c$

$f\left(x\right)=sinx$