Draw a graph to match the description given. Answers will...

Draw a graph to match the description given. Answers will vary.
$f(x)$ has a positive derivative over $(-\infty, 3)$ and (3,9) a negative derivative over $(9, \infty),$ and a derivative equal to 0 at $x=3$

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Key Concepts

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its variable. It provides information about how the function's output changes for a small change in the input.

Increasing Function

When the derivative is positive over an interval, it indicates that the function is increasing on that interval. This means that as the input increases, the output of the function also increases.

Decreasing Function

When the derivative is negative over an interval, it implies that the function is decreasing on that interval. This means that as the input increases, the output of the function decreases.

Critical Point

A critical point is a point on the function where the derivative is zero or undefined. When the derivative equals zero, it often indicates a potential local maximum, local minimum, or a point of inflection, depending on the behavior of the function around that point.

Graphical Representation Using Derivative Information

By analyzing where the derivative is positive, negative, or zero, one can sketch the behavior of the function. This includes identifying intervals where the function is increasing or decreasing and marking critical points to indicate potential turning points or horizontal tangents.

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