Question

mark the critical points on the following graph.

Explanation:

Define critical points

A critical point of a continuous function \(f(x)\) occurs where its derivative is zero or undefined. On a smooth graph, these correspond to points with horizontal tangent lines, such as local peaks and local valleys.

Identify the local maximum

Looking at the graph, there is a local maximum (a peak) in the interval \([-5, -3]\). The peak of this curve occurs exactly at the coordinates:

$$ (-4, 16) $$

Identify the local minimum

Looking further along the graph, the curve decreases, crosses the x-axis, and reaches a local minimum (a valley) in the interval \([0, 2]\). This valley occurs exactly at the coordinates:

$$ (1, -2) $$

Verify other regions

For \(x > 1\), the function increases strictly and rapidly towards infinity. There are no other peaks, valleys, or sharp corners where the derivative would be zero or undefined. Thus, the only critical points are the local maximum and local minimum identified.

Answer:

The critical points on the graph are located at:

• \((-4, 16)\) (local maximum)
• \((1, -2)\) (local minimum)