Tangents and it...

The line $$y = mx + 1$$ is a tangent to the curve $$y^{2} = 4x$$ if the value of m is .......

The slope of the normal to the curve $$ y = 2x ^{2} + 3 \sin x $$ at $$ x = 0 $$ is 

The line $$ y = x + 1 $$ is a tangent to the curve $$y^{2} = 4 x $$ at the point 

The points on the curve $$9 y^{2} = x^{3}$$, where the normal to the curve makes equal intercepts with the axes are ...........

The coordinates of a point P(x, y) lying in the first quadrant of the ellipse $$\displaystyle \frac{x^{2}}{8}+\frac{y^{2}}{18}=1$$ so that the area of the triangle formed by the tangent at P and the axes is the smallest are

The greatest inclination between the tangents is

A curve passes through $$(2, 0)$$ and the slope of the tangent at any point $$(x, y)$$ is $$x^2 -2x$$ for all values of $$x$$. The point of minimum ordinate on the curve where $$x > 0$$ is $$(a, b)$$'

The value of $$x$$ at which tangent to the curve $$y=x^3-6x^2+9x+4,   0\leq x \leq 5$$ has maximum slope is

A function $$y=f(x)$$ has a second order derivative $$f''(x)=6(x-1)$$ .
If its graph passes through the point $$(2,1)$$ and at that point the tangent to the graph is $$y=3x-5$$, then the function is

Suppose $$a,b,c$$ are such that the curve $$y = ax^2 + bx + c$$ is tangent to $$y = 3x -3$$ at $$(1, 0)$$ and is also tangent to $$y = x + 1$$ at $$(3, 4)$$ then the value of $$(2a -b -4c)$$ equals