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Compactness and connectedness
The two properties that make theorems work: extreme values and intermediate values.
Compact (on the line: closed and bounded) guarantees a maximum is attained; connected guarantees the intermediate value theorem. Picture it: a closed interval versus an open one. The function 1/x has no maximum on (0, 1]. Think it: both are defined by open covers and separations, so they transfer to any space.
हल किया गया उदाहरण · critical points of x^3 - 3x
चरण दर चरण
- f(x) = x^{3} - 3 x
Critical points are where f′(x) = 0 or is undefined.
- f'(x) = 3 x^{2} - 3
Differentiate.
- x = -1, x = 1
Solve f′(x) = 0.
- f''(x) = 6 x
Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.
- f''(-1) = -6 \Rightarrow (-1, 2) \text{ is a local maximum}
- f''(1) = 6 \Rightarrow (1, -2) \text{ is a local minimum}
जवाब दिखाएँ
अब आप कोई प्रश्न चुनें, या अपना प्रश्न टाइप करें या बनाएं। हर चरण, एक चित्र, और उत्तर तब तक छिपा रहता है जब तक आप न पूछें।
एक स्वतंत्र वृत्तान्त प्रत्येक पाठ पर आगे कहता है, जो आपने पूरा किया है, आपके हल किये गये समस्याओं का रिकॉर्ड, और एक शिक्षक जिसके बारे में आप पूछ सकते हैं ।
साइन अप करें लॉगिनयहाँ उपयोग में संकेत
पूरी परिभाषा, चित्र, और इसमें हर अक्षर का अर्थ क्या है?
कैसे: Compactness and connectedness
- Critical points are where f′(x) = 0 or is undefined.
- Differentiate.
- Solve f′(x) = 0.
- Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.
लोग जो सवाल पूछते हैं
Why "a doughnut is a coffee cup"?
Each can be deformed into the other without cutting or gluing: one hole each. Topology studies exactly the properties such deformations preserve.
What is compactness for?
It is the property that makes "every sequence has a convergent subsequence" and "continuous functions attain their maximum" true. On the real line it means closed and bounded.
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