Let for $f(x,y) = kxy - x^2y - xy^3$ for $(x,y) \in \Bbb{R}^2$ where $k$ is a real constant.The directional derivative of $f$ at the point $(1,2)$ in the direction of the unit vector $u = (-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$ is $\frac{15}{\sqrt{2}}$.
I applied the formula for the directional derivate at a point along the direction $u = (-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$, that is $\nabla f . \hat{a} = \frac{15}{\sqrt{2}}$ to get the value of $k = 4$.
Next I have to determine the value of $f$ at a local minimum in the rectangular region $R : \{(x,y) \in \Bbb{R}^2 : |x| < 1.5 , |y|<1.5\}$
I think we have a concept related to the maximum value that is "maximum value is attained on the boundary"