First Derivative Test for a Local Maximum

Question

In an experiment involving the analysis of a function, a graph is provided showing a smooth curve with a distinct local maximum at $$x=1$$. What would happen if a detailed derivative analysis were conducted in the neighborhood of $$x=1$$?

Accepted Answer

The derivative would change sign from positive to negative at $$x=1$$, confirming it as a local maximum.

Answer

The second derivative at $$x=1$$ would be zero, suggesting that $$x=1$$ is an inflection point rather than an extreme value.

Answer

The derivative would remain positive near $$x=1$$, indicating a constant increase in the function.

Answer

The function would exhibit a vertical asymptote at $$x=1$$ due to a sudden change in slope.