Solve the given problem by using substitution.
Find the maximum value of $f ( x , y , z ) = 3 - x ^ { 2 } - x - y ^ { 2 } + y - z ^ { 2 } + z$ subject to $y = 3 x$ .

A) $f _ { \max } = 3.225$
B) $f _ { \max } = 3$
C) $f _ { \max } = 3.35$
D) $f _ { \max } = 2.775$
E) $f _ { \max } = 2.45$

Question

Solve the given problem by using substitution.  
Find the maximum value of $f ( x , y , z ) = 3 - x ^ { 2 } - x - y ^ { 2 } + y - z ^ { 2 } + z$ subject to $y = 3 x$ .
 
A) $f _ { \max } = 3.225$
B) $f _ { \max } = 3$
C) $f _ { \max } = 3.35$
D) $f _ { \max } = 2.775$
E) $f _ { \max } = 2.45$

Quizplus · Accepted Answer

To find the maximum value of $f(x, y, z) = 3 - x^2 - x - y^2 + y - z^2 + z$ subject to $y = 3x$, we substitute $y$ with $3x$ in the function. This gives us $f(x, 3x, z) = 3 - x^2 - x - (3x)^2 + 3x - z^2 + z$. Simplifying, we get $f(x, z) = 3 - x^2 - x - 9x^2 + 3x - z^2 + z = 3 - 10x^2 + 2x - z^2 + z$. To find the maximum value, we take the partial derivatives with respect to $x$ and $z$, set them to zero, and solve for $x$ and $z$. The partial derivative with respect to $x$ is $-20x + 2 = 0$, giving $x = 0.1$. The partial derivative with respect to $z$ is $-2z + 1 = 0$, giving $z = 0.5$. Substituting these values back into the function gives $f(0.1, 0.5) = 3 - 10(0.1)^2 + 2(0.1) - (0.5)^2 + 0.5 = 3.35$, which is the maximum value.

Solve the Given Problem by Using Substitution f(x,y,z)=3−x2−x−y2+y−z2+zf ( x , y , z ) = 3 - x ^ { 2 } - x - y ^ { 2 } + y - z ^ { 2 } + zf(x,y,z)=3−x2−x−y2+y−z2+z

Solve the Given Problem by Using Substitution $f ( x , y , z ) = 3 - x ^ { 2 } - x - y ^ { 2 } + y - z ^ { 2 } + z$