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Chapter 4: Problem 70

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$f(x)=2 x^{4}+8 x^{3}+12 x^{2}-x-2$$

Short Answer

Expert verified
Question: Find the intervals of concavity and inflection points for the function $$f(x) = 2x^4 + 8x^3 + 12x^2 - x - 2$$. Answer: To find the intervals of concavity and inflection points, follow these steps: 1. Find the second derivative of the function. 2. Determine the critical points. 3. Analyze the intervals around the critical points. 4. Identify the inflection points. After going through these steps for the given function, provide the intervals of concavity and the inflection points (x-values).

Step by step solution

01

Find the second derivative

First, find the derivative of the function: $$f'(x) = \frac{d}{dx}(2x^4 + 8x^3 + 12x^2 - x - 2)$$ Now, take the second derivative: $$f''(x) = \frac{d^2}{dx^2}(f'(x))$$
02

Determine critical points

Find the critical points by setting the second derivative equal to zero and solve for x: $$f''(x) = 0$$
03

Analyze the intervals around the critical points

Divide the number line into intervals using the critical points found in step 2. For each interval, test a number in the second derivative to determine if the function is concave up or concave down.
04

Identify inflection points

An inflection point occurs where the concavity changes. Using the information from step 3, determine where the concavity changes from concave up to concave down or vice versa. The inflection points are the x-values of these points.

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