sketch the graph of a function f that is continuous on 1, 8 and has the given properties. absolute maximum…

Explanation:

Step1: Recall function - property requirements

The function is continuous on [1,8], has an absolute maximum at x = 3, an absolute minimum at x = 8, and x = 7 is a critical number with no local - max/min.

Step2: Analyze absolute maximum

The function value at x = 3 should be the highest among all function values on [1,8].

Step3: Analyze absolute minimum

The function value at x = 8 should be the lowest among all function values on [1,8].

Step4: Analyze critical number at x = 7

A critical number with no local max/min means the function has a horizontal tangent (or a non - differentiable point with a horizontal "kink") at x = 7 but does not change from increasing to decreasing or vice - versa at that point.

Step5: Evaluate the graphs

For the absolute maximum at 3, the function should peak at x = 3. For the absolute minimum at 8, it should be at the lowest point at x = 8. And at x = 7, the slope of the tangent line should be 0 (or have a non - differentiable horizontal feature) without a local extremum.

The correct graph is the one where the function reaches its highest point at x = 3, its lowest point at x = 8, and at x = 7, the function has a horizontal tangent (or a non - differentiable horizontal behavior) but does not have a local maximum or minimum. Without seeing the specific options clearly labeled, a general description of the correct graph would be: A continuous curve that starts at x = 1, increases to a peak at x = 3, then decreases, has a horizontal tangent (or non - differentiable horizontal feature) at x = 7 without changing from increasing to decreasing or vice - versa at that point, and then continues to decrease until it reaches its lowest point at x = 8.

If we assume the graphs are labeled as Option A, Option B, Option C, and Option D:

Answer:

Identify the graph that has a highest point at x = 3, a lowest point at x = 8, and a horizontal tangent (or non - differentiable horizontal feature) at x = 7 without a local maximum or minimum at x = 7.