<p data-start="0" data-end="182">Il Teorema di Lagrange, noto anche come <strong data-start="44" data-end="72">Teorema del Valore Medio, &egrave; un risultato fondamentale dell'analisi matematica che riguarda le funzioni derivabili. Esso afferma che:

<p data-start="184" data-end="369"><em data-start="184" data-end="367">Se una funzione <span class="katex"><span class="katex-mathml">f(x)f(x)</span><span class="katex-html"><span class="base"><span class="mord mathnormal">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> &egrave; continua in un intervallo chiuso <span class="katex"><span class="katex-mathml">[a,b][a, b]</span><span class="katex-html"><span class="base"><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> e derivabile nell'intervallo aperto <span class="katex"><span class="katex-mathml">(a,b)(a, b)</span><span class="katex-html"><span class="base"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span>, allora esiste almeno un punto <span class="katex"><span class="katex-mathml">c&isin;(a,b)c \in (a, b)</span><span class="katex-html"><span class="base"><span class="mord mathnormal">c</span><span class="mrel">&isin;</span></span><span class="base"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span> tale che:

<span class="katex-display"><span class="katex"><span class="katex-mathml">f&prime;(c)=f(b)&minus;f(a)b&minus;af'(c) = \frac{f(b) - f(a)}{b - a}</span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathnormal">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">&prime;</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mrel">=</span></span><span class="base"><span class="mord"><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="mord"><span class="mord mathnormal">b</span><span class="mbin">&minus;</span><span class="mord mathnormal">a</span></span><span class="mord"><span class="mord mathnormal">f</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mbin">&minus;</span><span class="mord mathnormal">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span><span class="vlist-s">​</span></span></span></span></span></span></span></span></span>

<h3 data-start="412" data-end="444">Interpretazione Geometrica</h3>


<p data-start="445" data-end="646">Il teorema dice che esiste almeno un punto <span class="katex"><span class="katex-mathml">cc</span><span class="katex-html"><span class="base"><span class="mord mathnormal">c</span></span></span></span> nell&rsquo;intervallo in cui la tangente al grafico della funzione &egrave; parallela alla retta secante che congiunge i punti <span class="katex"><span class="katex-mathml">(a,f(a))(a, f(a))</span><span class="katex-html"><span class="base"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mord mathnormal">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">))</span></span></span></span> e <span class="katex"><span class="katex-mathml">(b,f(b))(b, f(b))</span><span class="katex-html"><span class="base"><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mord mathnormal">f</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mclose">))</span></span></span></span>.