Application of ...

Consider the curve represented parametrically by the equation
$$\displaystyle x = t^3 - 4t^2 - 3t$$ and $$\displaystyle y = 2t^2 + 3t - 5$$ where $$\displaystyle t \: \epsilon \: R$$.
If $$H$$ denotes the number of point on the curve where the tangent is horizontal and $$V$$ the number of point where the tangent is vertical then

For a $$\displaystyle a\epsilon \left [ \pi , 2\pi  \right ],$$ the function $$\displaystyle f\left ( x \right )= \frac{1}{3}\sin a \tan ^{3}x+\left ( \sin a-1 \right )\tan x+ \frac{\sqrt{a-2}}{\sqrt{8-a}}$$

The point(s) on the curve $$y^{3}+3x^{2}=12y$$ the tangent is vertical is (are)

The coordinates of the point $$P$$ on the curve $$y^{2}= 2x^{3}$$ the tangent at which is perpendicular to the line $$4x-3y + 2 = 0$$, are given by

Find the co-ordinates of the point (s) on the curve $$\displaystyle y= \frac{x^{2}-1}{x^{2}+1}, x> 0$$ such that tangent at these point (s)have the greatest slope.

The tangent to the curve $$x= a\sqrt{\cos 2\theta }\cos \theta $$, $$y= a\sqrt{\cos 2\theta }\sin \theta
$$ at the point corresponding to $$\theta = \pi /6$$ is

The equation of the tangents to $$\displaystyle 4x^{2}-9y^{2}=36$$ which are perpendicular to the straight line $$\displaystyle 2y+5x= 10$$ are

The tangent to the curve $$y=e^{x}$$ drawn at the point $$\left ( c,e^{c} \right )$$ intersects the line joining the points $$(c -1,e^{c-1})$$ and $$(c +1,e^{c+1}) $$

For the curve $${x}^{2}+4xy+8{y}^{2}=64$$ the tangents are parallel to the $$x$$-axis only at the points

y = $$[x(x-3)]^2$$ is increasing when-