Answer
There is a critical point at $\left( {0, - \frac{1}{2},\frac{1}{4}} \right)$.
The critical value is $f\left( {0, - \frac{1}{2},\frac{1}{4}} \right) = \frac{1}{4}$.
However, this value does not correspond to the maximum nor minimum values of $f$ along the constraint.
Work Step by Step
We have $f\left( {x,y,z} \right) = {x^2} - y - z$ and the constraint $g\left( {x,y,z} \right) = {x^2} - {y^2} + z = 0$.
Step 1. Write out the Lagrange equations
Using Theorem 1, the Lagrange condition $\nabla f = \lambda \nabla g$ yields
$\left( {2x, - 1, - 1} \right) = \lambda \left( {2x, - 2y,1} \right)$
$2x = 2\lambda x$, ${\ \ }$ $ - 1 = - 2\lambda y$, ${\ \ }$ $ - 1 = \lambda $
Step 2. Solve for $\lambda$ in terms of $x$ and $y$
From Step 1, we obtain $\lambda = - 1$.
Step 3. Solve for $x$ and $y$ using the constraint
Substituting $\lambda = - 1$ in the equations in Step 1, we obtain $x=0$, $y = - \frac{1}{2}$. Substituting $x=0$ and $y = - \frac{1}{2}$ in the constraint $g\left( {x,y,z} \right)$ gives
$ - {\left( { - \frac{1}{2}} \right)^2} + z = 0$
$z = \frac{1}{4}$
So, the critical point (the red point in the figure attached) is $\left( {0, - \frac{1}{2},\frac{1}{4}} \right)$.
Step 4. Calculate the critical values
The value of $f$ at the critical point is $f\left( {0, - \frac{1}{2},\frac{1}{4}} \right) = \frac{1}{4}$.
Referring to the figure attached, we see that the level surface of $f$ whose value is $\frac{1}{4}$, intersects the constraint $g\left( {x,y,z} \right) = 0$ at the critical point $\left( {0, - \frac{1}{2},\frac{1}{4}} \right)$. However, we also notice that there are level surfaces of $f$ that have smaller values than $\frac{1}{4}$ and there are level surfaces of $f$ that have larger values than $\frac{1}{4}$ intersect the constraint $g\left( {x,y,z} \right) = 0$. Thus, the level surface whose value is $\frac{1}{4}$ is neither maximum nor minimum of $f$ along the constraint $g\left( {x,y,z} \right) = 0$. So, we conclude that $f\left( {0, - \frac{1}{2},\frac{1}{4}} \right) = \frac{1}{4}$ does not correspond to the maximum nor minimum values of $f$ along the constraint.
