Provide an Appropriate Response $t$ Corresponding to the Extrema Of $f ( x , y , z ) = \sin \left( x ^ { 2 } + y ^ { 2 } \right) \cos ( z )$

question 243

Multiple Choice

Provide an appropriate response.
-Find the value(s) of $t$ corresponding to the extrema of $f ( x , y , z ) = \sin \left( x ^ { 2 } + y ^ { 2 } \right) \cos ( z )$ subject to the constraints $x ^ { 2 } +$ $y ^ { 2 } = 6 t , 0 \leq t \leq \pi$ , and $z = \frac { \pi } { 6 }$ . Classify each extremum as a minimum or maximum. (Hint: $w = f ( x , y , z )$ is a differentiable function of t.

• A) \(t = \frac { \pi } { 12 }\), minimum; \(t = 0\), maximum
• B) \(t = \frac { \pi } { 12 }\), minimum
• C) \( \mathrm { t } = - \frac { \pi } { 12 }\), maximum
• D) \(\mathrm { t } = \frac { \pi } { 12 }\), minimum; \(\mathrm { t } = - \frac { \pi } { 12 }\), maximum

---

Definitions:

Respiratory Distress Syndrome

A serious condition affecting newborns causing breathing difficulties due to underdeveloped lungs and lack of a surfactant.

Apgar Scores

A quick test performed on a newborn at 1 and 5 minutes after birth, rating five criteria on a scale from 0 to 2, assessing the baby's overall health and determining the need for further medical intervention.

State of Arousal

A physiological and psychological condition of being alert and responsive to stimuli, which can vary in intensity.

Irregular Breathing

Breathing that varies in pattern and rhythm, which can be an indicator of health issues or distress.