Question

The graph of f', the derivative of the continuous function f, is shown above on the interval -3

Answer 1

A) f has three relative extrema, and the graph of f has one point of inflection.

Explanation:

On a graph of f'(x), relative extrema appear as points where the curve crosses the x-axis, indicating locations where the original function f(x) may have a local maximum or minimum. The sign of f'(x) changes from positive to negative at a relative maximum and from negative to positive at a relative minimum.

Therefore, according to the provided graph, f(x) has three relative extrema, as the graph of f'(x) crosses the x-axis at three points.

• Relative maximums at x = -2 and x = 5.
• Relative minimum at x = 0.

If the graph of f′(x) touches the x-axis but doesn't cross it, it suggests a point where the derivative is equal to zero but there is neither a relative maximum nor a relative minimum. This situation could indicate a point of inflection in the original function f(x). At a point of inflection, the function changes concavity without having a local extremum.

Therefore, according to the provided graph, f(x) has one point of inflection, as the graph of f'(x) touches the x-axis without crossing it at one point.