Given the polynomial function $ p(x) = x^3 - 6x^2 + 11x - 6 $, find the intervals where the function is increasing. - 1.4 Polynomial Functions and Rates of Change - Review - Use this assignment in your class!

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1.4 Polynomial Functions and Rates of Change - Review

Explain how the degree of a polynomial function affects its end behavior and rate of change.

Describe how the leading coefficient of a polynomial function influences its rate of change and overall graph shape.

How does the degree of a polynomial function influence the number of turning points it can have?

Consider the polynomial function $f (x) = - 2 x^{4} + 3 x^{3} - x + 5$ . Discuss how the leading coefficient and degree of this polynomial influence its graph.

How does the leading coefficient test help in determining the end behavior of a polynomial function?

Given the polynomial function $f (x) = 2 x^{3} - 3 x^{2} + x - 5$ , calculate the average rate of change of the function from $x = 1$ to $x = 4$ .

For the polynomial function $g (x) = x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$ , determine the instantaneous rate of change at $x = 2$ .

Consider the polynomial function $h (x) = 3 x^{2} - 2 x + 1$ . Calculate the average rate of change of $h (x)$ from $x = - 1$ to $x = 2$ .

Given the polynomial function $p (x) = x^{3} - 6 x^{2} + 11 x - 6$ , find the intervals where the function is increasing.

10.

For the polynomial function $q (x) = - 2 x^{3} + 3 x^{2} + 6 x - 1$ , determine the average rate of change from $x = - 2$ to $x = 1$ .

11.

Given the polynomial function $f (x) = 2 x^{3} - 3 x^{2} + x - 5$ , determine the intervals where the function is increasing and decreasing. Explain your reasoning.

12.

Consider the polynomial function $g (x) = - x^{4} + 4 x^{3} - 6 x^{2} + 4 x$ . Determine the local maxima and minima of the function.

13.

Analyze the end behavior of the polynomial function $h (x) = 3 x^{5} - 2 x^{3} + x - 7$ . Describe what happens to $h (x)$ as $x$ approaches $\infty$ and $- \infty$ .

14.

For the polynomial function $p (x) = x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$ , determine the concavity and points of inflection.

15.

Given the polynomial function $q (x) = x^{3} - 6 x^{2} + 9 x + 15$ , find the x-coordinates of the points where the function has a horizontal tangent line.

Given the polynomial function $p (x) = x^{3} - 6 x^{2} + 11 x - 6$ , find the intervals where the function is increasing.

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AI InstructorAUG 11, 2024, 3:07:27 PM

To find the intervals where $p (x)$ is increasing, we need to find the derivative $p^{'} (x)$ and determine where it is positive:
$p^{'} (x) = \frac{d}{d x} (x^{3} - 6 x^{2} + 11 x - 6) = 3 x^{2} - 12 x + 11$
We solve $3 x^{2} - 12 x + 11 > 0$ . First, find the critical points by solving $3 x^{2} - 12 x + 11 = 0$ :
$x = \frac{12 \pm \sqrt{144 - 132}}{6} = \frac{12 \pm 2 \sqrt{3}}{6} = 2 \pm \frac{\sqrt{3}}{3}$
The critical points are $x = 2 + \frac{\sqrt{3}}{3}$ and $x = 2 - \frac{\sqrt{3}}{3}$ . Test intervals around these points to determine where $p^{'} (x)$ is positive. The function is increasing on the intervals $(- \infty, 2 - \frac{\sqrt{3}}{3})$ and $(2 + \frac{\sqrt{3}}{3}, \infty)$ .

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On which intervals is the function $p (x) = x^{3} - 6 x^{2} + 11 x - 6$ increasing?

Select one of the following options:

A.
$(- \infty, 2 - \frac{\sqrt{3}}{3})$ and $(2 + \frac{\sqrt{3}}{3}, \infty)$
B.
$(- \infty, 2 + \frac{\sqrt{3}}{3})$
C.
$(2 - \frac{\sqrt{3}}{3}, 2 + \frac{\sqrt{3}}{3})$
D.
$(2 - \frac{\sqrt{3}}{3}, \infty)$

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Given the polynomial function p(x)=x3−6x2+11x−6, find the intervals where the function is increasing.

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On which intervals is the function p(x)=x3−6x2+11x−6 increasing?

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Given the polynomial function $p (x) = x^{3} - 6 x^{2} + 11 x - 6$ , find the intervals where the function is increasing.

On which intervals is the function $p (x) = x^{3} - 6 x^{2} + 11 x - 6$ increasing?