नीचे दिए गए फलनों की भूयिष्ठकों अथवा अल्पिष्ठकों हेतु जांच कीजिए:

(क) z= 2x² + xy + 4y² +xz + 2² +2

(ख)

आपने जो सवाल पूछा है, उसमें z = 2x² + xy + 4y² + xz + 2x + 2y को लेकर भूयिष्ठक (partial derivatives) और अल्पिष्ठक (second-order partial derivatives) की जांच करने के लिए कहा गया है। हम इसे एक-एक करके हल करेंगे।

1. भूयिष्ठक (First-Order Partial Derivatives)

यहां हम पहले z का x और y के अनुसार भूतार्थिक अवकलन करेंगे। फिर, x, y, और z के अनुसार दूसरे-क्रम के अवकलन करेंगे।

### (क) z = 2x² + xy + 4y² + xz + 2x + 2y

x के संदर्भ में भूयिष्ठक (∂z/∂x):

z को x के संदर्भ में अवकलित करने के लिए हम x की शक्ति बढ़ाने या घटाने के हिसाब से प्रत्येक पद का अवकलन करेंगे।

- 2x² का अवकलन x के संदर्भ में: 4x
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