6. Graphical Applications of Derivatives

Intro to Extrema

6. Graphical Applications of Derivatives

Intro to Extrema

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Multiple Choice

Determine where the local and absolute maxima and minima occur on the given graph of $f\left(x\right)$ .

Absolute max of 5 at $x=4$ , Local max of 4 at $x=-3$ , Absolute min of $-2$ at $x=1$ , Local min of $-2$ at $x=1$ & 1.5 at $x=-4.5$

Absolute max of 5 at $x=4$ , Absolute min of $-2$ at $x=1$ , No local extrema

Absolute max of 4 at $x=-3$ , Local max of 4 at $x=-3$ , Absolute min of $-2$ at $x=1$ , Local min of $-2$ at $x=1$

Absolute max of 5 at $x=4$ , Local max of 4 at $x=-3$ , Absolute min of $-2$ at $x=1$ , Local min of $-2$ at $x=1$ $x=1$

Verified step by step guidance

Step 1: Observe the graph of f(x) and identify the points where the function reaches its highest and lowest values. These are the absolute maxima and minima.

Step 2: Look for peaks and valleys in the graph to identify local maxima and minima. A local maximum occurs at a peak, and a local minimum occurs at a valley.

Step 3: From the graph, note that the absolute maximum value of 5 occurs at x=4, as this is the highest point on the graph.

Step 4: Identify the local maximum value of 4 at x=-3, as this is a peak that is not the highest point overall.

Step 5: Observe that the absolute minimum value of -2 occurs at x=1, and this is also a local minimum. Additionally, there is another local minimum value of 1.5 at x=-4.5.