Question

Let $f$ be a function satisfying $f"\left(x\right)=x^{-3/2},f^{\prime }\left(4\right)=2$ and $f\left(0\right)=0.$ Then $f\left(784\right)$ is equal to

Answer 1

$f^{\prime \prime }\left(x\right)=x^{-3/2}$
Integrate w.r.t $x$
$f^{\prime }\left(x\right)=-2x^{-1/2}+c$
As $f^{\prime }\left(4\right)=2\Rightarrow -2.4^{-1/2}+c=2\Rightarrow c=3$
$f^{\prime }\left(x\right)=-2x^{-1/2}+3$
Integrating w.r.t $x$
$f\left(x\right)=-4x^{1/2}+3x+df\left(0\right)=0\Rightarrow d=0f\left(x\right)=-4x^{1/2}+3xf\left(784\right)=-4{\times 784}^{1/2}+4\times 784=2240$