Number of Intercepts and Local Extrema (a) How many x -intercepts and how many local extr

step 1 All right, so I've gone ahead and graphed out both A and B here. A is the blue graph. B is the g

Number of Intercepts and Local Extrema (a) How many x ...

Number of Intercepts and Local Extrema \begin{array}{l}{\text { (a) How many } x \text { -intercepts and how many local extrema }} \\ {\text { does the polynomial } P(x)=x^{3}-4 x \text { have? }}\end{array} \begin{array}{l}{\text { (b) How many } x \text { -intercepts and how many local extrema }} \\ {\text { does the polynomial } Q(x)=x^{3}+4 x \text { have? }} \\ {\text { (c) If } a>0 \text { , how many } x \text { -intercepts and how many local }} \\ {\text { extrema does each of the polynomials } P(x)=x^{3}-a x} \\ {\quad \text { and } Q(x)=x^{3}+a x \text { have? Explain your answer. }}\end{array}

Number of Intercepts and Local Extrema

\begin{array}{l}{\text { (a) How many } x \text { -intercepts and how many local extrema }} \\ {\text { does the polynomial } P(x)=x^{3}-4 x \text { have? }}\end{array}
\begin{array}{l}{\text { (b) How many } x \text { -intercepts and how many local extrema }} \\ {\text { does the polynomial } Q(x)=x^{3}+4 x \text { have? }} \\ {\text { (c) If } a>0 \text { , how many } x \text { -intercepts and how many local }} \\ {\text { extrema does each of the polynomials } P(x)=x^{3}-a x} \\ {\quad \text { and } Q(x)=x^{3}+a x \text { have? Explain your answer. }}\end{array}

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Key Concepts

Derivative and Critical Points

The derivative of a function provides information about the function’s rate of change. By setting the derivative equal to zero, one can find the critical points, which are potential candidates for local extrema, and determine the function’s increasing and decreasing intervals.

Local Extrema

Local extrema refer to the local maximum and minimum points of a function. They represent the highest or lowest points in a particular region of the graph and are found by analyzing the critical points where the derivative is zero or undefined.

Polynomial Functions

Polynomial functions are expressions involving sums of powers of x with constant coefficients. They are continuous and differentiable everywhere, and their behavior at infinity is determined by the leading term, which helps in predicting the end behavior of the graph.

X-intercepts

X-intercepts (or zeros) are the points where the graph of a polynomial crosses or touches the x-axis. Finding these intercepts involves solving for values of x that make the polynomial equal to zero, often by factoring or using other algebraic methods.

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