F^'(x) is given in factored form, and p(x) is a function...

$ f^{\prime}(x)$ is given in factored form, and $p(x)$ is a function that is always positive. Find all critical values of $f,$ the largest open intervals on which $f$ is increasing. the largest open intervals on which $f$ is decreasing, and all relative maxima and all relative minima. Using the additional information given for each function, sketch a rough graph of $f$ $f^{\prime}(x)=p(x)(x-2)(x-4), f(2)=2, f(4)=-2$

Key Concepts

Critical Points

Critical points are the x-values where the derivative of a function equals zero or is undefined. In this context, since the derivative is given in factored form and multiplied by a positive function, the zeros of the derivative (from the factors that can change sign) are the critical points. These points are candidates for local maxima and minima.

First Derivative Test

The first derivative test involves examining the sign changes of the derivative around the critical points to determine whether each is a local maximum, a local minimum, or neither. By observing the behavior of the derivative (from positive to negative or vice versa) as the function crosses each critical point, one can classify the extrema.

Sign Chart Analysis

A sign chart is used to evaluate the sign of the derivative on the intervals determined by the critical points. This analysis identifies where the function is increasing (positive derivative) or decreasing (negative derivative). When a factor in the derivative is always positive, the sign changes are dictated solely by the other factors.

Intervals of Increase and Decrease

The intervals on which a function is increasing are those where the first derivative is positive, and the intervals where the function is decreasing are those where the derivative is negative. Determining these intervals is essential in understanding the overall behavior of the function.

Relative Extrema

Relative extrema refer to the local maximum and minimum values of a function. By applying the first derivative test, one can identify if a critical point is a relative maximum (where the function transitions from increasing to decreasing) or a relative minimum (where it transitions from decreasing to increasing).

Graph Sketching Based on Derivative Information

Sketching a rough graph of the function involves combining the information about the critical points, intervals of increase and decrease, and known function values. This holistic approach allows one to visualize the behavior of the function by starting with its derivative and then incorporating specific function evaluations.

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