Q4Advanced Engineering Mathematics

We are tasked with finding the extremum values of the objective function $Z=x^2+y^2+z^2$ subject to a specific equality constraint $4x+y^2+2z=14$. This is a classic constrained optimization problem that is best solved using the method of Lagrange Multipliers.

First, we rewrite the constraint equation in the standard form $g(x,y,z)=0$: $g(x,y,z)=4x+y^2+2z-14=0$

1. Formulating the Lagrangian Function

We construct the Lagrangian function $L(x,y,z,\lambda )$ by combining the objective function and the constraint multiplied by the Lagrange multiplier $\lambda$:

2. Finding the Stationary Points

To find the stationary points, we must compute the partial derivatives of $L$ with respect to $x,y,z,$ and $\lambda$, and rigidly set them all to zero. This creates a system of four simultaneous equations:

From Equation 2 ($2y(1+\lambda )=0$), we arrive at two distinct mathematical possibilities to satisfy the condition: either $y=0$ or $\lambda =-1$. We must rigorously analyze both cases to find all potential extrema.

Case 1: Assume $y=0$

If $y=0$, we substitute $x=-2\lambda$ and $z=-\lambda$ directly into the constraint Equation 4:

Now, substituting $\lambda =-1.4$ back into our expressions for $x$ and $z$:

$x=-2(-1.4)=2.8$ and $z=-(-1.4)=1.4$.

This gives us our first stationary point: $(2.8,0,1.4)$. Let us calculate the value of the objective function $Z$ at this specific coordinate:

Case 2: Assume $\lambda =-1$

If $\lambda =-1$, we immediately determine the values of $x$ and $z$ from Equations 1 and 3:

$x=-2(-1)=2$ and $z=-(-1)=1$.

Now, we substitute $x=2$ and $z=1$ into the constraint Equation 4 to solve for the remaining variable $y$:

This yields two additional stationary points: $(2,2,1)$ and $(2,-2,1)$. We calculate the objective function $Z$ for these points:

3. Conclusion

We compare the calculated values of $Z$ from our different cases. We found $Z_1=9.8$ and $Z_2=9.0$. Comparing these, the absolute minimum value is clearly 9. Therefore, the constrained optimal minimum value of the function is $Z=9$, which uniquely occurs at the coordinate points $(2,2,1)$ and $(2,-2,1)$.