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Understanding Critical Points in Graphs: Maxima, Minima, and Inflection Points

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This guide explains critical points in functions and their significance in graph analysis. A maximum occurs when a graph increases to the left and decreases to the right of a point (top of a hill), while a minimum is where it decreases on the left and increases on the right (bottom of a valley). We also cover points of inflection, where curvature changes, and define relative and absolute extrema. Instructions for using a calculator to find these points in function graphs are included, alongside examples to practice.

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Understanding Critical Points in Graphs: Maxima, Minima, and Inflection Points

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  1. 3.6 Critical Points

  2. Critical Points .Maximum: When the graph is increasing to the left of x = c and decreasing to the right of x = c (top of hill) Minimum: When the graph of a function is decreasing to the left of x = c and increasing ot the right of x = c (bottom of valley) Point of Inflection: a point where the graph changes its curvature.

  3. Extremum – a minimum or maximum value of a function • Relative Extremum– a point that represents the maximum or minimum for a certain interval • Absolute Maximum – the greatest value that a function assumes over its domain • Relative Maximum – a point that represents the maximum for a certain interval (highest point compared to neighbors ) • Absolute Minimum – the least value that a function assumes over its domain • Relative Minimum – a point that represents the minimum for a certain interval (minimum compared to neighbourhors0

  4. To find a point in the calculator • Use your best estimate to locate a point • 2nd – calc- max/min • Place curser on left, enter. Place curser on right, enter. Enter

  5. Graph the following examples and pick out the critical points • F(x) = 5x3 -10x2 – 20x + 7 • F(x) = 2x5 -5x4 –10x3.

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