Suppose that f'(c)=0, f''(c)=0 and f'''(c)>0 . What, if anything, happens to the graph of the function f at x = c? Explain clearly and completely.

the slope has a minimum at x=c.

f"(c)=0 means it's an inflection point.

Think of the graph f(x) = x^3

So is it just an inflection point?

yes. The f"' nonzero makes sure that it's not just a cup like x^4.

To understand what happens to the graph of the function f at x = c when f'(c)=0, f''(c)=0, and f'''(c)>0, let's break down the information step by step:

1. f'(c)=0: This tells us that at x = c, the function f has a critical point, meaning its slope is zero at that point. There could be various possible scenarios regarding the behavior of the graph at x = c.

2. f''(c)=0: This indicates that at x = c, the function f has a point of inflection. A point of inflection is a point where the concavity of the function changes.

3. f'''(c)>0: This condition tells us that the rate of change of the slope of the function is positive at x = c. In other words, the curvature of the graph at x = c is concave up.

Considering these conditions, the following possibilities arise:

1. The graph of f could have a local minimum at x = c if the function changes from a negative slope to a positive slope, indicating a concave up behavior.

2. The graph of f could have a local maximum at x = c if the function changes from a positive slope to a negative slope, indicating a concave down behavior.

3. The graph could also have neither a local minimum nor a local maximum at x = c if the function is either linear or has a point of inflection where the concavity changes but the function doesn't exhibit extremum behavior.

To determine which of these possibilities occurs, further analysis of the function and its behavior around x = c is necessary. The first derivative test and the second derivative test can be employed to investigate the behavior and nature of the critical point at x = c, specifically to determine if it corresponds to a minimum, maximum, or neither.

It's important to keep in mind that the given conditions f'(c)=0, f''(c)=0, and f'''(c)>0 provide information about the local behavior of the function at x = c, but do not necessarily give a definitive answer about the presence of local extrema. Additional analysis and information may be required to make a conclusive determination.