Application of ...

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

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

If the normal to the curve $$\displaystyle y= f\left ( x \right )$$ at the point $$\displaystyle \left ( 3, 4 \right )$$ makes an angle $$\displaystyle \frac{3\pi}{4}$$ with the positive x-axis then $$\displaystyle f'\left ( 3 \right )$$ is equal to

Find the points on the curve $$y=x^{3}$$, the tangents at which are inclined at an angle of $$60^{\circ}$$ to x-axis.

Find the condition that the line $$\displaystyle Ax+By= 1$$ may be a normal to the curve $$\displaystyle a^{n-1}y=x^{n}.$$

If the normal to the curve $$y=f(x)$$ at the point $$(3,4) $$ makes an angle $$3\pi /4 $$ with the positive x-axis, then $$f'(3)=$$

The set of all values of x for which the function $$\displaystyle f\left ( x \right )= \left ( k^{2}-3k+2 \right )\left ( \cos ^{2}\frac{x}{4}-\sin ^{2}\frac{x}{4} \right )+\left ( k-1 \right )x+\sin 1$$ does not posses critical points is 

Find the co-ordinates of the points on the curve $$\displaystyle y= x/\left ( 1+x^{2} \right )$$ where the tangent to the curve has greatest slope.

Find $$\displaystyle \frac{dy}{dx}$$ if$$ \:y= \left [ x+\sqrt{x+} \sqrt{x}\right ]^{1/2}$$, at $$x=1$$

A and B are points $$(-2,0)$$ and $$(1,3)$$ on the curve $$\displaystyle y=4-x^{2}$$. If the tangent at P on the curve be parallel to chord AB, then co-ordinates of point P are