Chapter 8.3, Problem 41E

To determine

Whether the statement “Suppose $h\left(x,y\right)=f\left(x\right)+g\left(y\right)$, where f and g have continuous second derivatives near a and b, respectively. If a is a critical number of f, b is the critical number of g, and $f"\left(a\right)g"\left(b\right)>0$ , then h has a relative extremum at $\left(a,b\right)$ is true or false. Explain the reason if it is true or given an example if it is false.