Classifying the stationary point of $h(x,y,z) = 2(x−1)^2 + 3(y−1)^3 + 4(z−1)^4$

Question

Given the function $h(x,y,z) = 2(x−1)^2 + 3(y−1)^3 + 4(z−1)^4$, I have found the only stationary point to be $(1,1,1)$. I then attempted to use the Hessian matrix to find out whether $(1,1,1)$ is a local minimum/maximum/saddle point, however this was inconclusive.

Is there another way I can go about classifying this point?

Any help would be very appreciated! Thanks.

Answer 1

We have $ h(1,1,1)=0$ and $h(1,y,1) =3(y-1)^3$

Hence

$$h(1,y,1) >0 =h(1,1,1)$$

for $y>1$ and

$$h(1,y,1) <0 =h(1,1,1)$$

for $y<1$ .

Conclusion ?