Question

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consider (f(x)=x^{3}-5x^{2}+3x + 2). without graphing the function, answer each question.

1. name the y - intercept.
2. how many zeros are there? determine the consecutive integer values of x between which each real zero is located.
3. how many turning points are there? estimate the locations of the turning points.
4. describe the end behavior.

Explanation:

Step1: Find y - intercept

Set \(x = 0\) in \(f(x)=x^{3}-5x^{2}+3x + 2\). Then \(f(0)=0^{3}-5\times0^{2}+3\times0 + 2=2\).

Step2: Determine number of zeros

The degree of the polynomial \(f(x)=x^{3}-5x^{2}+3x + 2\) is \(n = 3\), so the maximum number of real - zeros is 3. Use the Intermediate Value Theorem. \(f(-1)=(-1)^{3}-5\times(-1)^{2}+3\times(-1)+2=-1 - 5-3 + 2=-7\), \(f(0)=2\), \(f(1)=1^{3}-5\times1^{2}+3\times1 + 2=1-5 + 3+2=1\), \(f(2)=2^{3}-5\times2^{2}+3\times2 + 2=8-20 + 6+2=-4\), \(f(3)=3^{3}-5\times3^{2}+3\times3 + 2=27-45 + 9+2=-7\), \(f(4)=4^{3}-5\times4^{2}+3\times4 + 2=64-80 + 12+2=-2\), \(f(5)=5^{3}-5\times5^{2}+3\times5 + 2=125-125+15 + 2=17\). There are 3 real zeros, and they are located between \(-1\) and \(0\), \(1\) and \(2\), \(4\) and \(5\).

Step3: Find number of turning points

The degree of the polynomial is \(n = 3\), so the number of turning points is at most \(n - 1=2\). First, find the derivative \(f^\prime(x)=3x^{2}-10x + 3\). Set \(f^\prime(x)=0\), then \(3x^{2}-10x + 3 = 0\), factoring gives \((3x - 1)(x - 3)=0\), so \(x=\frac{1}{3}\) and \(x = 3\) are the approximate locations of the turning points.

Step4: Describe end - behavior

Since the leading coefficient of \(f(x)=x^{3}-5x^{2}+3x + 2\) is \(a = 1>0\) and the degree \(n = 3\) (odd), as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow+\infty\), \(y
ightarrow+\infty\).

Answer:

1. The \(y\) - intercept is 2.
2. There are 3 zeros. They are located between \(-1\) and \(0\), \(1\) and \(2\), \(4\) and \(5\).
3. There are at most 2 turning points, located approximately at \(x=\frac{1}{3}\) and \(x = 3\).
4. As \(x

ightarrow-\infty\), \(y
ightarrow-\infty\); as \(x
ightarrow+\infty\), \(y
ightarrow+\infty\).