The graph of the derivative f^' of a function f is shown....

The graph of the derivative $f^{\prime}$ of a function $f$ is shown.
(a) On what intervals is $f$ increasing? Decreasing?
(b) At what values of $x$ does $f$ have a local maximum? Local minimum?
(c) If it is known that $f(0)=0$, sketch a possible graph of $f.$
(GRAPH CAN'T COPY)

Key Concepts

Antiderivatives and Graph Sketching

Using the derivative information, one can sketch the original function by considering the accumulation of the derivative (i.e., finding the antiderivative) along with given initial conditions. This process involves understanding how the slopes indicated by the derivative translate into the shape and curvature of the original function graph.

Local Extrema and the First Derivative Test

Local maximums and minimums of a function can often be identified by examining where the derivative zeroes out and changes sign. If the derivative changes from positive to negative, the function likely has a local maximum, and if it changes from negative to positive, a local minimum is indicated. This procedure, known as the first derivative test, is a fundamental method in calculus for detecting local extrema.

Derivative

The derivative of a function represents the instantaneous rate of change or slope of the function at any given point. It provides the essential connection between the function and its behavior, indicating how rapidly the function values are changing, which is crucial for understanding the dynamic behavior of the function.

Function Monotonicity

The sign of the derivative indicates whether the function is increasing or decreasing. When the derivative is positive, the function is increasing; when it is negative, the function is decreasing. This concept helps in determining the intervals over which the function rises or falls.

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