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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.

ምሳሌ · critical points of x^3 - 3x

Calculus · Extrema

Critical points of x^3 - 3x

x^{3} - 3 x

እርምጃ በደረጃ

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}

መልሱን አሳይ

(-1, 2)\ \text{local maximum},\; (1, -2)\ \text{local minimum}

አሁን ችግርን ይምረጡ፣ ወይም ጻፉ ወይም በራሳችሁ ቅርጽ አቅርቡ። እርምጃው ሁሉ፣ ምስል፣ መልሱ እስከምትጠይቅ ድረስ የተደበቀ ነው።

የራስዎን ስራዎች ይያዙ

ነጻ የሆኑት መተግበሪያዎች በየክፍል ውስጥ ማስታወሻዎችን ያካትታሉ፣ የተጠናቀቁትን ነገሮች መዝገብ፣ የተፈቱ ችግሮችን በአንድ ቦታ፣ ስለዚህ ገጽ መጠየቅ የሚችሉትን ረዳት ያካትታሉ፡፡ የቁጥር ትምህርት ለሁሉም ሰው ክፍት ነው፣ ተቀባይነት ያለው ወይም የለም፤

ምዝገባ መግባቱ

ፊደላት

ለሙሉ መግለጫ፣ ምስል፣ እና በውስጡ ያለውን ፊደል ትርጓሜ ለማየት ማንኛውንም ምልክት ይጫኑ።

እንዴት: 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.

ጥያቄዎች

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.

በ Topology

Metric spaces and open sets Homeomorphism and topological invariants