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 $f$ $f^{\prime}(x)=p(x)(x+1)^{2}(x-1)^{3}(x-2), f(-1)=-2,$ $f(1)=3, f(2)=0$

$ 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 $f$
$f^{\prime}(x)=p(x)(x+1)^{2}(x-1)^{3}(x-2), f(-1)=-2,$
$f(1)=3, f(2)=0$

Key Concepts

Multiplicity of Roots in the Derivative

The multiplicity of a factor in the derivative's factored form affects how the sign of the derivative changes around the corresponding critical point. An even multiplicity means the sign does not change as the graph touches and rebounds off the x-axis, while an odd multiplicity indicates a sign change, affecting the determination of relative extrema.

Relative Extrema

Relative (or local) extrema occur at critical points where the function changes its direction of monotonicity. The first derivative test is used to classify these extremes: if the derivative changes from positive to negative at a critical point, the function has a local maximum there, and if it changes from negative to positive, there is a local minimum.

Graph Sketching of Functions

Sketching the graph of a function based on derivative information involves using the identified critical points and intervals of monotonicity, along with any given function values. This process integrates information about increasing or decreasing behaviour, relative extrema, and the overall shape of the graph to provide a rough outline of the function’s behavior.

Critical Points

Critical points of a function are the x-values where its derivative is zero or undefined. They are important because they indicate where the function might change its rate of increase or decrease, and therefore may be locations of local extrema (either minima or maxima).

Intervals of Monotonicity

Determining where a function is increasing or decreasing involves examining the sign of its derivative. When the derivative is positive, the function is increasing; when it is negative, the function is decreasing. Constructing a sign chart for the derivative can help identify these intervals along the domain of the function.

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