Question

A function $y=f\left(x\right)$ satisfying $f"\left(x\right)=x^{-3/2},f^{\prime }\left(4\right)=2$ and $f\left(0\right)=0$ is $-\lambda \sqrt{x}+3x$, then find the value of $\lambda$

Answer 1

$\frac{d^{2}y}{d{x}^2}=x^{-3/2}$
$\frac{dy}{dx}=\frac{x^{-1/2}}{-1/2}+c$

$\because \frac{dy}{dx}atx=4=-1+c\Rightarrow 2=c-1\Rightarrow c=3$

$\frac{dy}{dx}=-\frac{2}{\sqrt{x}}+3$

$y=-2\frac{x^{1/2}}{\frac{1}{2}}+3x+d$
$\because yatx=0=d\Rightarrow 0=d$
$y=-4\sqrt{x}+3x$