A. How many x -intercepts and how many local extrema does...

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

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

Key Concepts

Polynomial Functions

Polynomial functions, particularly cubic functions, are expressions where the highest power of x is three. Their end behavior is determined by the leading coefficient, and they can have up to three real roots and up to two turning points. Understanding these functions is crucial to analyzing their graphs and behavior across the domain.

X-intercepts

X-intercepts are the points where the graph of a polynomial function crosses the x-axis; these are the real solutions to the equation P(x)=0. The number of x-intercepts of a polynomial is determined by its degree and the nature of its roots. In the context of cubic functions, factorization often reveals the existence of one or three real roots, indicating the number of x-intercepts.

Local Extrema

Local extrema refer to the points on a graph where the function reaches a local maximum or minimum. For polynomial functions, these extrema are found by examining critical points, which are determined by setting the first derivative equal to zero. The number of local extrema in a cubic function can vary depending on the coefficients of the function, with derivatives having distinct real solutions indicating the presence of turning points.

Transcript

00:02 So in this question, in the first part we have y is equal to px that is given to me as x cube minus 4x.

00:11 So if i observe the graph of this one, i will simplify this one first x into x square minus 4.

00:17 So i can see that it has roots x equal to 0 and plus minus 2.

00:24 So the graph will gradually come like this.

00:27 It's a simple cubic, y -axis, x -axis, a simple cubic nature, 2 .0, minus 2 .0, 2 .0.

00:38 So clearly i can see x intercept are 3 x intercepts and local extrema if i talk about it has one maxima and one minima.

00:57 So clearly this is the maxima, this is the minima.

01:01 Now in the second part i have a polynomial qx given to me as x cube plus 4x.

01:09 So again about the intercepts i will have this polynomial as x taken common.

01:14 X squared plus 4 so clearly only one roots are there okay so root roots are only x equal to 0 other roots are imaginary over here also if i check the maxima or minima it will have no maxima no minima all right so i will say this function has or this polynomial has only one x intercept x intercept and no extrema because it is a continuously increasing function without any point where the derivative is equal to 0.

02:05 In the third part i have a polynomial px given to me as x cube minus a x.

02:11 So if i observe this polynomial carefully, they have mentioned to me that a value is greater than 0.

02:20 So this is px and one more polynomial is given to me as qx, that is x cube plus ax.

02:29 Here also a is greater than 0...

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