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Chapter 2: Problem 39

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$f(x)=3 x^{3}-10 x+9 ; \text { between }-3 \text { and }-2$$

Short Answer

Expert verified
Yes, according to the Intermediate Value Theorem, the polynomial \(f(x)=3x^{3}-10x+9\) has at least one real root between -3 and -2.

Step by step solution

01

Evaluate the polynomial at the endpoints of the interval

Calculate \(f(-3)\) and \(f(-2)\).
02

Determine if the signs differ

If the signs of \(f(-3)\) and \(f(-2)\) are different, then there is a root between -3 and -2.
03

Calculation

Now calculate \(f(-3)=3*(-3)^3-10*(-3)+9=-9\). Then, calculate \(f(-2)=3*(-2)^3 -10*(-2)+9=1.\) The signs of these two numbers are different; therefore, the Intermediate Value Theorem guarantees that there is at least one real root of the given polynomial between -3 and -2.

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Most popular questions from this chapter

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=1,\) a slant asymptote whose equation is \(y=x, y\) -intercept at \(2,\) and \(x\)-intercepts at -1 and 2.

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=3,\) a horizontal asymptote \(y=0, y\) -intercept at \(-1,\) and no \(x\) -intercept.

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

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