5. 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 $x=1$

Verified step by step guidance

Identify the highest and lowest points on the graph. The highest point is the absolute maximum, and the lowest point is the absolute minimum.

Observe the graph to find the absolute maximum. The point at which the graph reaches its highest value is at x=4, where f(x)=5.

Look for the absolute minimum on the graph. The lowest point on the graph is at x=1, where f(x)=-2.

Determine the local maxima by finding peaks that are not the absolute maximum. A local maximum occurs at x=-3, where f(x)=4.

Identify the local minima by finding valleys that are not the absolute minimum. A local minimum occurs at x=1, where f(x)=-2.

Master Finding Extrema Graphically with a bite sized video explanation from Patrick

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Expert video explanations

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