Question

mark the critical points on the following graph.
clear all draw: dot

Explanation:

Step1: Recall the definition of critical points

Critical points of a function \(y = f(x)\) are the points where \(f^{\prime}(x)=0\) (horizontal tangent) or \(f^{\prime}(x)\) does not exist. On a graph, these are the points where the slope of the tangent line is zero (peaks, valleys, and flat - regions).

Step2: Identify the peaks, valleys, and flat - regions on the graph

• The valley (local minimum) occurs where the function changes from decreasing to increasing.
• The peak (local maximum) occurs where the function changes from increasing to decreasing.
• The flat - region (where the function has a horizontal tangent and is neither increasing nor decreasing in a small neighborhood) also counts as a critical point.

Looking at the given graph:

• There is a local minimum (valley) in the left - hand side of the graph.
• There is a flat - region between \(x=-1\) and \(x = 1\) (where the slope of the tangent line is \(0\)).
• There is a local maximum (peak) near \(x = 4\).

Answer:

Mark the local minimum (valley), the flat - region (between \(x=-1\) and \(x = 1\)), and the local maximum (peak) on the graph.