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<p>Import words via API</p> <p><b>New page</b></p><div>==== valsi ====<br /> cnansari<br /> ==== type ====<br /> fu&#039;ivla<br /> ==== creator ====<br /> [[personal/krtisfranks|krtisfranks]]<br /> ==== time entered ====<br /> Sun Apr 1 05:01:25 2018<br />  <br /> == English ==<br /> === Definition #70583 - Preferred ===<br />  <br /> ==== definition ====<br /> x&lt;sub&gt;1&lt;/sub&gt; is the mean-value theorem mean/forward-difference-quotient mean of the elements of (multi)set x&lt;sub&gt;2&lt;/sub&gt; (1-element or 2-element set) under/for function x&lt;sub&gt;3&lt;/sub&gt;.<br /> ==== notes ====<br /> <br /> Let the sumti of &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt;&lt;/span&gt; be a real-valued, univariate function &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;f&lt;/i&gt;&lt;/span&gt; which is defined and continuous on at least the closed interval [ MATH<br /> $min(x_2), max(x_2)$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;min&lt;/i&gt;(&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;), &lt;i&gt;max&lt;/i&gt;(&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;)&lt;/span&gt;] and differentiable on at least the open interval MATH<br /> $(min(x_2), max(x_2))$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;(&lt;i&gt;min&lt;/i&gt;(&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;), &lt;i&gt;max&lt;/i&gt;(&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;))&lt;/span&gt;, such that the derivative &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;f&#039;&lt;/i&gt;&lt;/span&gt; is injective. Then MATH<br /> $x_1 = (f&#039;)^&lt;a class=&quot;undefined&quot; href=&quot;../dict/(-1)?bg=1;langidarg=2&quot;&gt;(-1)&lt;/a&gt; ((f(max(x_2)) - f(min(x_2)))/(max(x_2)-min(x_2)))$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt; = (&lt;i&gt;f&#039;&lt;/i&gt;)&lt;sup&gt;(-1)&lt;/sup&gt;((&lt;i&gt;f&lt;/i&gt; (&lt;i&gt;max&lt;/i&gt;(&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;)) - &lt;i&gt;f&lt;/i&gt; (&lt;i&gt;min&lt;/i&gt;(&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;)))/(&lt;i&gt;max&lt;/i&gt;(&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;) - &lt;i&gt;min&lt;/i&gt;(&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;)))&lt;/span&gt; if &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/span&gt; has cardinality 2; if &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/span&gt; has cardinality 1, then &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;&lt;/span&gt; is equal to the sole element of that set; other repairs may be necessary, depending on &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;f&lt;/i&gt;&lt;/span&gt;: for example, &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;f&lt;/i&gt; = &lt;i&gt;log&lt;/i&gt;&lt;/span&gt; has &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt; = 0&lt;/span&gt; if 0 is an element of &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/span&gt;. If &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt;&lt;/span&gt; is filled with the monic, single-term polynomial of (integer) degree &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;p&lt;/i&gt;&lt;/span&gt; such that &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;p&lt;/i&gt; &amp;gt; 1&lt;/span&gt;, then &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;&lt;/span&gt; will be the principal (p-1)th root of the (1/p)-scaled weighted (p,1)-Lehmer mean ([[cnanlime|cnanlime]]) of the elements of &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/span&gt; wherein the weight of the lesser of them is -1 (and the other weight is +1) unless &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/span&gt; is a singleton (in which case the weights are both +1). Usually, &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;&lt;/span&gt; is the value which is guaranteed to exist for appropriate function &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/span&gt; by the Mean-Value Theorem.<br /> <br /> ==== gloss words ====<br /> <br /> &lt;ul&gt;&lt;li&gt;[[natlang/en/forward-difference-quotient mean|forward-difference-quotient mean]]&lt;/li&gt;&lt;li&gt;[[natlang/en/logarithm mean|logarithm mean]]&lt;/li&gt;&lt;li&gt;[[natlang/en/log mean|log mean]]&lt;/li&gt;&lt;li&gt;[[natlang/en/MVT mean|MVT mean ; Mean-Value Theorem mean]]&lt;/li&gt;&lt;/ul&gt;<br /> <br /> ==== created by ====<br /> [[personal/krtisfranks|krtisfranks]]<br /> ==== vote information ====<br /> 1<br /> <br /> ==== time ====<br /> Sun Apr 1 05:59:29 2018<br /> <br /> <br /> <br /> &lt;br/&gt;&lt;font size=&quot;+1&quot;&gt;Examples&lt;/font&gt;<br /> &lt;hr/&gt;<br /> &lt;dl&gt;<br /> &lt;/dl&gt;<br /> <br /> === Etymology ===</div>

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