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- As <math>F</math> is concave, its derivative <math>F'</math> is monotonically decreasing. We consider two cases.3 KB (628 words) - 05:29, 28 November 2024
- A function <math>f:A\to B</math> is called [[monotonically increasing]] if <math>f(x_1)\geq f(x_2) </math> holds whenever <math>x_1>x_ Similarly, a function <math>f:A\to B</math> is called [[monotonically decreasing]] if <math>f(x_1)\geq f(x_2) </math> holds whenever <math>x_1<x_2 </math>.10 KB (1,761 words) - 02:16, 12 May 2023
- ...monotonically decreasing on the [[interval]] <math>(-\infty, 0]</math> and monotonically increasing on the interval <math>[0, \infty)</math>. However, the function More formally, a function <math>f</math> is monotonically increasing (resp. decreasing) if <math>a \leq b \Longrightarrow f(a) \leq f(b)</math> (resp. <math>f(a)1 KB (155 words) - 16:15, 22 August 2006
- ...c{1}{\sqrt{x}}</math>. Note that this function is convex and monotonically decreasing which implies that if <math>a > b</math>, then <math>f(a) < f(b)</math>.3 KB (453 words) - 23:12, 27 March 2021
- ...e that <math>A\in [60^{\circ},120^{\circ}]</math>. Cosine is monotonically decreasing on this interval, so by setting <math>A</math> at the extreme values, we se4 KB (647 words) - 15:28, 1 September 2021
- ...math>and therefore the sequence <math>\{x_n\}</math> will be monotonically decreasing after one point. ...h> that maps <math>x_1</math> to <math>x_n</math>, which is continuous and monotonically increasing, with <math>\lim_{x\to +\infty} f_n(x)=+\infty</math> so <math>f9 KB (1,784 words) - 16:40, 15 October 2024
- ...itive, <math>f(x)</math> is also positive. Therefore, <math>f(x)</math> is decreasing on <math>[6, 8]</math>, and the maximum value occurs at <math>x = 6</math>. ...th> and it decreases to 0 at <math>x=8</math>. Thus, <math>p(x)</math> is decreasing over the entire domain of <math>f(x)</math> and it obtains its maximum va3 KB (491 words) - 21:21, 30 September 2021
- ...ath>0<x<1</math>, <math>f(n)=\log_x(n)</math> is [[monotonic|monotonically decreasing]], so the order of terms by magnitude in our new set of numbers will be rev ...ath>x<x^x</math>, and <math>f(n)=x^n</math> is a [[monotonic|monotonically decreasing]] function, we know that <math>x^x>x^{x^x}</math>. However, because <math>x2 KB (373 words) - 06:56, 18 July 2024
- Thus, <math>f(n) > f(n+1)</math>; i.e., <math>f</math> is monotonic decreasing. Therefore, because <math>f(0) > 0</math>, there exists a unique <math>N</m Therefore, <math>g</math> is also monotonic decreasing. Note that <math>g(N+1) = a_0 + a_1 + \dots + a_{N+1} - (N+1) a_{N+1} \le 03 KB (611 words) - 10:16, 8 July 2023
- ...union of two monotonically increasing subsequences (Case 1). Consider any monotonically increasing subsequence that starts at <math>x = a</math> and ends at <math> ...atisfied at <math>(x, y) = \left(x_1, x_3\right)</math>. If we ever have a decreasing part where <math>f(x + 1) < f(x)</math>, then we can use some variant of th9 KB (1,607 words) - 09:27, 4 February 2022