If the equation of tangent at $(2,3)$ on $y^2=a x^3+b$ is $y=4 x-5$, then the value of $a^2+b^2=$

If Rolle's theorem is applicable for the function $f(x)=x(x+3) e^{-x / 2}$ on $[3,0]$, then the value of $c$ is

For all $x \in[0,2024]$ assume that $f(x)$ is differentiable, $f(0)=-2$ and $f^{\prime}(x) \geq 5$. Then, the least possible value of $f(2024)$ is

A point is moving on the curve $y=x^3-3 x^2+2 x-1$ and the $y$-coordinate of the point is increasing at the rate d 6 units per second. When the point is at $(2,-1)$, the rate of change of $x$-coordinate of the point is