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118 lines
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118 lines
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<h1 align="center">Maxima and Minima of Functions</h1>
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<h2>Local Maximum and Minimum</h2>
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<p>Functions can have "hills and valleys": places where they reach a minimum or maximum value. </p>
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<p>It may not be the minimum or maximum for the <b>whole function</b>, but <b>locally</b> it is.</p>
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<p align="center"><img src="images/function-max-min.svg" alt="Local Max and Min"></p>
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<p align="center">We can see where they are, <br>
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but how do we define them?</p>
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<h2>Local Maximum</h2>
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<p><b>First</b> we need to choose an interval:</p>
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<p align="center"><img src="images/function-max-interval.svg" alt="Local Max in Interval"></p>
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<p>Then we can say that a local <b>maximum</b> is the point where:</p>
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<p>The height of the function at "a" is greater than (or equal to) the height anywhere else in that interval.</p>
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<p>Or, more briefly:</p>
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<p class="larger" align="center">f(a) ≥ f(x) for all x in the interval</p>
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<p>In other words, there is no height greater than f(a).</p>
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<p>Note: a should be <b>inside </b>the interval, not at one end or the other.</p>
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<h2>Local Minimum</h2>
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<p>Likewise, a local <b>minimum</b> is:</p>
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<p class="larger" align="center">f(a) ≤ f(x) for all x in the interval</p>
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<p>The plural of Maximum is <b>Maxima</b></p>
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<p>The plural of Minimum is <b>Minima</b></p>
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<p>Maxima and Minima are collectively called <b>Extrema </b></p>
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</div>
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<h2>Global (or Absolute) Maximum and Minimum</h2>
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<p>The maximum or minimum over the <b>entire function</b> is called an "Absolute" or "Global" maximum or minimum.</p>
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<p>There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum.</p>
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<p class="center"><img src="images/function-max-global.svg" alt="Global Maximum"></p>
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<p><b>Assuming</b> this function continues downwards to left or right:</p>
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<ul>
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<li>The Global Maximum is about 3.7</li>
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<li>The Global Minimum is −Infinity</li>
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</ul>
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<h2>Calculus</h2>
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<p><a href="../calculus/index.html">Calculus</a> can be used to find the exact <a href="../calculus/maxima-minima.html">maximum and minimum using derivatives</a>.</p>
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<p> </p>
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<div class="related"><a href="../calculus/maxima-minima.html">Maxima and Minima using Derivatives</a> <a href="index.html">Algebra Index</a></div>
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