Question
The curve given by $$x + y = {e^{xy}}$$ has a tangent parallel to the $$y$$-axis at the point :
A.
(0, 1)
B.
(1, 0)
C.
(1, 1)
D.
none of these
Answer :
(1, 0)
Solution :
$$\eqalign{
& {\text{Differentiating w}}{\text{.r}}{\text{.t}}{\text{. }}\,x,\,1 + \frac{{dy}}{{dx}} = {e^{xy}}\left( {y + x\frac{{dy}}{{dx}}} \right) \cr
& {\text{or }}\frac{{dy}}{{dx}} = \frac{{y{e^{xy}} - 1}}{{1 - x{e^{xy}}}} \cr
& \frac{{dy}}{{dx}} = \infty \,\,\,\,\,\,\,\, \Rightarrow 1 - x{e^{xy}} = 0\,\,\,\,\,\,\, \Rightarrow 1 - x\left( {x + y} \right) = 0 \cr
& {\text{This holds for }}x = 1,{\text{ }}y = 0. \cr} $$