Match the LIST-I with LIST-II

LIST-I Function $f(x)$	LIST-II $f(x)$ is increasing in the interval
A. $f(x)=-x^2-2x+3$	I. $(-\infty,2)$
B. $f(x)=x^2+4x+1$	II. $(1,\infty)$
C. $f(x)=2x^2-4x+3$	III. $(-\infty,-1)$
D. $f(x)=-3x^2+12x+1$	IV. $(-2,\infty)$

Choose the correct answer from the options given below:

For $x \in \mathbb{R}$, if $f(x) = -(x-1)^2 + 2$, then

(A) $f$ is an increasing function on $(-\infty, 1]$

(B) $f$ has no critical points

(C) $f$ has a maximum value at $x = 1$

(D) $f$ has a minimum value at $x = 1$

Choose the correct answer from the options given below:

If the total cost function is given by $C(x)=\frac{x^3}{3}-2x^2+9x-70$ and the selling price per unit is ₹6, then for what value of $x$ will the profit be maximum?

The rate of change of $\sqrt{x^2-2}$ with respect to $\dfrac{x}{x+1}$ at $x=2$ is :

In a sphere of radius $r$, a right circular cone of height $h$ having a maximum curved surface area is inscribed. The expression for the curved surface area of the cone is

The equation of the tangent line to the curve $y = x^2 - 2x + 5$ which is parallel to the line $4x - y + 1 = 0$ is

The point(s) on the curve $\frac{x^2}{9} + \frac{y^2}{64} = 1$, at which the tangents are parallel to x-axis are