Application of ...

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 ...........

For $$a\in[\pi,2\pi]$$ and $$n\in I$$, the critical points of $$\displaystyle f(x)=\frac{1}{3}\sin a\tan^{3}x+(\sin a - 1 ) \tan x +\sqrt{\frac{a-2}{8-a}}$$ is

The value of a for which the function $$\displaystyle \mathrm{f}(\mathrm{x})=(4\mathrm{a}-3)(\mathrm{x}+\log 5)+2(\mathrm{a}-7)\cot\frac{\mathrm{x}}{2}\sin^{2}\frac{\mathrm{x}}{2}$$ does not possess critical points is

The greatest inclination between the tangents 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

If $$ f(x) = \displaystyle \frac{x}{{\sin x}}$$ and $$g(x) = \displaystyle \frac{x}{{\tan x}}$$  where $$0 < x \leq 1$$ then in the interval

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 $$(a, b)$$'

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