The shortest distance between line $$y-x=1$$ and curve $$x = {y^2}$$ is

The equation of the tangent to the curve $$y = x + {4 \over {{x^2}}}$$, that
is parallel to the $$x$$-axis, is

Let $$f:R \to R$$ be defined by $$$f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} & {x \le - 1} \cr {2x + 3,\,\,if} & {x > - 1} \cr } } \right.$$$

If $$f$$has a local minimum at $$x=-1$$, then a possible value of $$k$$ is

Let $$f:R \to R$$ be a continuous function defined by $$$f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}$$$

Statement - 1 : $$f\left( c \right) = {1 \over 3},$$ for some $$c \in R$$.

Statement - 2 : $$0 < f\left( x \right) \le {1 \over {2\sqrt 2 }},$$ for all $$x \in R$$