Science:Math Exam Resources/Courses/MATH105/April 2016/Question 01 (d)/Solution 1

Suppose that $f(x,y)=x+y$ and that the domain of $f$ is $[0,1]^{2}$ (i.e., the set of points $(x,y)$ satisfying $0\leq x\leq 1$ and $0\leq y\leq 1$).

Since $0\leq x\leq 1$ and $0\leq y\leq 1$, clearly $f$ has an absolute minimum of $0+0=0$ at $(0,0)$ and an absolute maximum of $1+1=2$ at $(1,1)$.

However, neither $(0,0)$ nor $(1,1)$ is a critical point of $f$ because $\nabla f=(\partial _{x}f,\partial _{y}f)=(1,1)$ (recall that a critical point of $f$ is a point $(x,y)$ at which $\nabla f(x,y)=(0,0)$).

Therefore, the statement is true.