Application of ...

The line $$y=x$$ is a tangent to the parabola $$\displaystyle y= ax^{2}+bx+c$$ at the point $$x=1$$.If the parabola passes through the point $$(-1,0)$$, then determine $$a, b, c.$$

Find the points on the curve $$y=x/(1-x^{2})$$ where the tangents makes an angle of $$\pi /4$$ with x-axis

The line $$\dfrac xa+\dfrac yb=1$$ touches the curve $$\displaystyle y=be^{-x/a}$$ at the point

The curve $$\displaystyle y-e^{xy}+x=0$$ has a vertical tangent at the point 

If $$\displaystyle x\cos \alpha +y\sin \alpha =p$$ touches $$\displaystyle x^{2}+a^{2}y^{2}=a^{2},$$ then

The critical points of the function $$\displaystyle f\left ( x\right )=\left ( x-2 \right )^{2/3}\left ( 2x+1\right )$$ are

If $$\displaystyle f\left ( 0 \right )=0$$ and $$\displaystyle f''\left ( x \right )>0$$ for all $$x > 0$$, then $$\displaystyle \frac{f(x)}{x}$$

The number of critical points of the fuction $$\displaystyle f'\left ( x \right ),$$ where $$\displaystyle f'\left ( x \right )= \frac{\left | x-2 \right |}{x^{3}}$$ are

The angle at which the curve $$y=ke^{kx}$$ intersects the $$y$$ -axis is

Let $$\displaystyle f\left ( x \right )=x^{3}+ax+b$$ with $$\displaystyle a\neq b$$ and suppose the tangent lines to the graph of $$f$$ at $$x = a$$ and $$x = b$$ have the same gradient Then the value of $$f (1)$$ is equal to