Answer
Local maximum at $f(\dfrac{16}{7},0)=\dfrac{-16}{7}$
Work Step by Step
Given: $f_x(x,y)=\dfrac{112x-8x}{\sqrt{56x^2-8y^2-16x-31}}=0, f_y(x,y)=\dfrac{-8y}{\sqrt{56x^2-8y^2-16x-31}}=0$
Simplify the given two equations.
This implies that $x=\dfrac{16}{7},y=0$
Critical point: $(\dfrac{16}{7},0)$
In order to solve this problem we will have to apply Second derivative test that suggests the following conditions to calculate the local minimum, local maximum and saddle points of $f(x,y)$ or $f(x,y,z)$.
1. If $D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 \gt 0$ and $f_{xx}(a,b)\gt 0$ , then $f(a,b)$ is a local minimum.
2. If $D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 \gt 0$ and $f_{xx}(a,b)\lt 0$ , then $f(a,b)$ is a local maximum.
3. If $D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 \lt 0$ , then $f(a,b)$ is a not a local minimum or local maximum but a saddle point.
$D=f_{xx}f_{yy}-f^2_{xy}=\dfrac{64}{225}\implies D=\dfrac{64}{225} \gt 0$ and $f_{xx}=-\dfrac{-8}{15} \lt 0$
Hence, Local maximum at $f(\dfrac{16}{7},0)=\dfrac{-16}{7}$
