Use Lagrange multiplier to maximize $f\left( x,y \right) =x^3y^5$ subject to constraint $ g\left( x,y \right) =x+y=8$. We need to find all points $(x,y)$ that satisfy the condition: $$ \begin{align} \nabla f=\lambda \nabla g ,\\ \langle f_x\left( x,y \right) ,f_y\left( x,y \right) \rangle =\lambda \langle g_x\left( x,y \right) ,g_y\left( x,y \right) \text{)}\rangle ,\\ f_x\left( x,y \right) =3y^5x^2, f_y\left( x,y \right) =5x^3y^4 ,\\ f_x=\lambda g_x,f_y=\lambda g_y ,\\ f_x=\lambda ,f_y=\lambda ,\\ 3y^5x^2=\lambda ,5x^3y^4=\lambda ,\\ \end{align} $$
Given the constrain is $g\left( x,y \right) =x+y=8$.
Suppose $\lambda =0$ , then either $x$ or $y$ are equal to 0, but that contradicts equation $x+y=8$, it means the point does not lie on the constraint.
Question: may I know please why the point does not lie on the constraint?
Hint:
If $3x^2y^5=5x^3y^4$ then $3y=5x$ and so you have a system $$5x-3y=0$$ $$x+y=8$$ to solve.