Question

mark the critical points on the following graph.

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.

Step2: Identify horizontal tangent points on the graph

Looking at the graph, we can see the points where the slope of the tangent line is zero (horizontal). These are the local minima, local maxima, and points of inflection (if slope - change from non - zero to zero).

• The local minimum occurs at \(x=-4\) (where the function changes from decreasing to increasing).
• The local maximum occurs at \(x = 4\) (where the function changes from increasing to decreasing).
• There are also points at \(x=-1\) and \(x = 1\) where the slope of the tangent is zero (the function has a "flat" part at these \(x\) - values in the graph).

Answer:

Mark dots at \(x=-4\), \(x=-1\), \(x = 1\), and \(x = 4\) on the graph.