The graph of the derivative f^' of a continuous function f is shown. (a) On what intervals is f

The graph of the derivative f^' of a continuous function f...

The graph of the derivative $f^{\prime}$ of a continuous function $f$ is shown. (a) On what intervals is $f$ increasing or decreasing? (b) At what values of $x$ does $f$ have a local maximum or minimum? (c) On what intervals is $f$ concave upward or downward? (d) State the $x$ -coordinate(s) of the point(s) of inflection. (e) Assuming that $f(0)=0,$ sketch a graph of $f.$

The graph of the derivative $f^{\prime}$ of a continuous function $f$ is shown.
(a) On what intervals is $f$ increasing or decreasing?
(b) At what values of $x$ does $f$ have a local maximum or minimum?
(c) On what intervals is $f$ concave upward or downward?
(d) State the $x$ -coordinate(s) of the point(s) of inflection.
(e) Assuming that $f(0)=0,$ sketch a graph of $f.$

Key Concepts

Derivative and Monotonicity

The derivative of a function indicates the rate of change of the function. When the derivative is positive, the function is increasing; when it is negative, the function is decreasing. This concept is fundamental for determining intervals of increase or decrease by analyzing the sign of the first derivative.

Critical Points and Local Extrema

Critical points occur where the derivative is zero or undefined. These points are important in identifying local maxima or minima. A change in the sign of the derivative from positive to negative suggests a local maximum, while a change from negative to positive indicates a local minimum.

Concavity and Second Derivative

Concavity relates to how the function curves. A function is concave upward if its second derivative is positive, indicating that the slope is increasing, and concave downward if the second derivative is negative, meaning the slope is decreasing. Although the second derivative might not be provided explicitly, its behavior can be inferred from the graph of the first derivative.

Inflection Points

Inflection points are points on the graph of a function where the concavity changes; that is, the second derivative changes sign. Identifying these points is key to understanding the overall curvature of the function.

Sketching from Derivative

Sketching the original function from the graph of its derivative involves using the information about where the function is increasing, decreasing, its local extrema, and its concavity. Given an initial condition, one can integrate the behavior of the derivative to reconstruct an approximate graph of the original function.