Solve the Given Problem by Using Substitution $f ( x , y , z ) = 2 x ^ { 2 } + 2 x + y ^ { 2 } - y + z ^ { 2 } - z - 4$

question 9

Multiple Choice

Solve the given problem by using substitution.
Find the minimum value of $f ( x , y , z ) = 2 x ^ { 2 } + 2 x + y ^ { 2 } - y + z ^ { 2 } - z - 4$ subject to $z = 2 y$ .

• A) \(f _ { \min } = - 3.775\)
• B) \(f _ { \min } = - 3.5\)
• C) \(f _ { \min } = - 4\)
• D) \(f _ { \min } = - 4.95\)
• E) \(f _ { \min } = - 4.7\)

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Definitions:

Ball Sockets

A type of bearing that allows for rotational movement in all directions, commonly used in automotive steering systems.

Ackerman Arms

Components of the steering mechanism that help achieve different turning radii for the inner and outer wheels, improving cornering.

Pitman Arm

A steering linkage arm that connects a steering gear to a drag link. A Pitman arm is usually splined to the steering gear sector shaft. In a typical steering gear, the Pitman arm converts the rotary motion of the sector shaft into linear motion at the drag link plus adds leverage.

Steering Control Arm

The steering control arm is a vital component of the steering system, connecting the steering knuckle to the vehicle's frame or body, thus allowing for effective steering wheel control over the vehicle's direction.