5. Graphical Applications of Derivatives

Intro to Extrema

5. Graphical Applications of Derivatives

Intro to Extrema

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

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

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

Absolute max of 4 at $x=-2$ , Local max of 4 at $x=-2$ , No absolute or local mins

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

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

Verified step by step guidance

Examine the graph to identify any peaks or valleys, which represent local maxima or minima. A local maximum is a point where the function changes from increasing to decreasing, and a local minimum is where it changes from decreasing to increasing.

Observe the graph at x = -2. The graph reaches a peak at this point, indicating a local maximum. The value of the function at this point is 4.

Check the endpoints of the graph to determine absolute maxima or minima. The graph starts at x = -5 with a value of 2 and continues to decrease beyond x = 5.

Since the graph does not rise above the value of 4 at x = -2, this point is also the absolute maximum.

The graph does not have any points where it changes from decreasing to increasing, so there are no local minima. The lowest point on the graph is at x = 5, but it is not a minimum since the graph continues to decrease.